TOTAL
395
Algebra r of Operations P lease Parenthesis - Do all grouped operations first. E xcuse Exponents - Second M y Multiplication and Division - Left to Right. D ear A unt Addition and Subtraction - Left to Right. S haniqua Follow the correct order of operations to evaluate expressions. Evaluate: Remember to use the correct Order of Operations. 1. 2 5 18 2. 2 6 12 2 3. 10 2 3 2 2 Evaluate for a=3, b=4, c=5, d=10 1. d bc ab 2. b a ad c 3. 2 2 a c bd Solve the following using the correct order of operations: 1. 3 3 3 3 6. 3 3 3 3 2 2. 3 3 3 3 7. 3 3 3 3 3. 3 3 ) 3 3 ( 8. 3 ) 3 3 3 ( 4. 3 3 3 3 9. 3 3 3 3 5. ) 3 3 ( ) 3 3 ( 10. 3 3 3 3 2 1.2 Order of Operations
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Algebrar of OperationsP lease Parenthesis - Do all grouped operations first.
E xcuse Exponents - Second
M y Multiplication and Division - Left to Right.D ear
A unt Addition and Subtraction - Left to Right.S haniqua
Follow the correct order of operations to evaluate expressions.
Evaluate: Remember to use the correct Order of Operations.
1. 2518 2. 26122 3.
10232 2
Evaluate for a=3, b=4, c=5, d=10
1. dbcab 2. baadc
3. 22 acbd
Solve the following using the correct order of operations:
1. 3333 6. 33332
2. 3333 7. 3333
3. 33)33( 8. 3)333(
4. 3333 9. 3333
5. )33()33( 10. 33332
1.2Order of Operations
AlgebraWhen evaluating expressions, work using the correct order of operations:
P (Parenthesis) Do all grouped operations first.E (Exponents) Do all operations involving exponents.M D (Mult./Div.) Do all multiplication and division from left to right.A S (Add./Sub.) Do all addition and subtraction last - from left to right.
Solve:
1. 5)19( 2 2. 2315
3. )25)(36( 4. )]29(4[2
5. 2)19(6 6. 3
)25( 2
7. 5
362 8. 3237 2
9. 6)315(43
10. 22 )335(
83
Order of Operations PracticeName________________________ Period _____
1.2
AlgebraEvaluate for a=3, b=4, c=6
11. 2)2( bac 12. 22 ab
13. 2))(( bcba 14. )]3([2 abc
15. )( 2 abc 16. a
bc 2)(6
17. 3abc
18. acb 2
19. cacb 3)( 20. caba 2
Order of Operations PracticeName________________________ Period _____
1.2
AlgebraInteger Addition 1.4notes:
Integers are positive and negative Whole Numbers like
-9 127 -90 -54 75 120 65 21 -78 -23 -11 70
Integers are NOT decimals or fractions.
Adding and subtracting integers can seem unnecessarily complicated.Try the following practice problems first:
Practice:
1. 3113 2. 3113 3. 1331 4. )31(13
5. 1331 6. )13(31 7. 1331 8. )13(31
If you got all of these right, you already have a proven method for addingand subtracting integers. Close your ears, sit quietly, and continue usingyour own method. If you missed even one, pay close attention and takenotes.
notes:Adding Integers:
Same Sign SumWhen adding integers with the same sign, find the sum and keep the
sign of both numbers.
1. 1113 2. )11(13 3. 223 4. )2(23
Different Sign DifferenceWhen adding integers with different signs, find the difference and
keep the sign of the ‘bigger’ number.
1. 1113 2. )11(13 3. )2(23 4. 223
Mixed ReviewAdd:
1. )14(15 2. )8(3 3. )8(7 4. )6(13
5. )14(12 6. 116 7. )5(9 8. )23(23
AlgebraSubtraction 1.4Subtracting Integers:
SMATO Subtraction Means Add The Opposite
Subtracting Integers is more complicated than adding integers.To subtract integers, change subtraction to addition and switch the sign
of the second number. Then, follow the two rules we have learnedfor adding integers.
Examples: SMATOChange to addition.
1. 1511 2. 321 3. )14(8 4. )5(30
Practice: Change to addition, then solve.
1. 2513 4. )6(29
2. 1511 5. 2315
3. )26(17 6. )7(2129
Adding and Subtracting Rationals:Use the same rules for fractions and decimals as you would for integers:Same Sign Sum, Different Sign Difference, SMATO.
Examples:
1. 101
41 2. 9.45.3 3.
87
31
4. )75.2(25.4 5. 215
611 6. 03.04.1
Practice:
1. 32
21 2. 5.49.1 3.
109
54
4. )1.2(2.6 5. 415
213 6. )05.1(9.2
AlgebraMatrices: A matrix is a rectangular table of numbers.Horizontal lines are called rows. Vertical lines are called columns.
953
32
1A
325
710
B
17
30
15
12
C
Matrix A and B are 3x2 matrices. C is a 2x4 matrix.Q: In matrix C, which number is in the second row, third column?
Matrix addition/subtraction.To add A+B, simply add the corresponding parts.You can only add or subtract matrices with the same dimensions.Subtraction is easier if you simply add the opposite.
953
32
1A
325
710
B
1238
101
1BA
672
43
1BA
Practice: Solve the following using the given matrices:
4
352
41
A
41
30
19
B
17
52
01
C
1. A+B 2. A-C 3. B-C 4. A+C
Matrices 1.5+
AlgebraMatrix multiplication: Multiplying a matrix by a scalar (a numbershown outside the matrix) involves multiplying each term by the scalar.
953
32
1A
1810
6
64
22A
22
6
028
113
01
42
Complete the following operations involving matrices:
52
41
A
63
25
B
14
04
C
1. A2 2. B2 3. BA 4. CB 3
Solving Matrix Equations:Ex:
95 x 372 x
One-step. Solve for B: Two-steps. Solve for A:
0
312
52
41
B
13
54
73
70
2A
Practice:Solve for the missing matrix in each problem below:
1.
174
523
953
32
1A
2.
374
103
396
115
2A
Matrices 1.5+
AlgebraComplete the following problems using the given matrices:Write ‘impossible’ if a problem cannot be solved.
953
32
1A
325
710
B
17
30
15
12
C
1. BA
2. A3
3. CC 2
4. AC
5. BA 2
6. Solve for matrix X:
341
5613
751
241
X
Matrix OperationsName________________________ Period _____
1.5+
AlgebraCombining Like Terms:When adding or subtracting numbers and variables, you can only
combine like terms.
L i k e t e r m s contain the same variables, with the same exponentsin a single product.
Here are some sets of like terms:
xx
x
35
2
2
2
39
xyxy
xy
ababab
23
nmnm
nm
3
3
3
22
6
cc
c
52
Practice: Match each pair or set of like terms below:
3x xy7 yx22 2xy yx3
3y yx 2 x3 22y 225 yxxy5 22 yx 32x xy9 yx 24
You cannot add or subtract unlike terms. It is like trying to add applesand oranges.
Practice: Simplify the following:
1. xx 79 2. zz 213
3. aaa 65 4. 22 53 mmmm
5. xx 15423 6. abbcb 3792 32
You cannot simplify number six because there are NO LIKE TERMS.
Combining Like Terms 2.3
AlgebraPractice: Simplify.
1. aa 314 2. ababab 728
3. ccc 1234 4. 22 201538 xxxx
5. 125247 xx 6. 361535 32 bbbb
Combining Like Terms 2.3
Practice: Simplify.
1. aa31
41
2.
xx
32
53
3. 22
51 aa 4. aba
32
21
98 2
5. 22
21
31
41 xyyxxy 6. bbb
21
52
Practice: Simplify.
1. yaxyax 53 2. abababab 735
3. cbca 7234 4. 22 101583 yxxyyx
5. yxyx 2547 22 6. 222
43
32
21 bba
AlgebraLike Terms Reteach:Name________________________ Period _____
Combining like terms is just like adding and subtracting integers: Simplify.
1. 73 2. 94 3. 61
4. xx 73 5. aa 94 6. xyxy 6
7. 55 73 xx 8. 33 94 xyxy 9. 2525 6 yxyx
You can only combine terms with the same variables and exponents: Simplify.Write SIMPLIFIED if there are no terms which can be combined. Circle like termsas you combine them.
7. 73 aa 8. xx 945
9. 2332 63 yxxy 10. baabab 2222 263
11. 3232 xxxx 12. 2332 327 mnnm
13. 32332 265 yyyyy 14. 2222 32 wxwwxxww
2.3
AlgebraLike Terms Reteach:Name________________________ Period _____
Fraction review: Solve.
15. 32
21 16. 3
21212 17.
543
212
18. 32
21 19. 3
21212 20.
543
212
Simplify each by combining like terms.
21. 732
21
aa 22. abab 1072
81
23. yxxyyx 333 332
24. 222
43
47 aaa
25. 322223 5
527
31 mmnnmmnnmmnmnm
2.3
AlgebraLike Terms PracticeName________________________ Period _____
Simplify each. Write simplified if no terms can be combined.
1. 3232 352 xxxx 2. 2332 3472 mnnm
3. 3232 235 xyxyxyxy 4. 2332 3232 ababaa
5. 2222 52 wxwxwxwx 6. 62 2332 baba
7. 735 33 yzxyzx 8. 5252 2 abababab
9. 2232 33 yxyxyx 10. 2332 2552 mmmm
11. 3232 37 mcemce 12. xaaxaxxa 2332 10205
13. 222
522
51 xxx 14. bababa 222
52
32
43
2.3
AlgebraBase:
The repeated factor in a power.In the expression n³, n is the base.
Exponent:Represents the number of times a factor is being multiplied.In the expression n³, the ³ is the exponent.
The expression 35 means that you multiply 555
The expression 5x means that you multiply xxxxx
The expression 4)(ab means that you multiply ))()()(( abababab
Practice: Write-out without using exponents:
1. 53 2. 4xy 3.
32 )( ba 4. 23)5( yx
Practice: Write using exponents.
1. xxx 777 2. bbaaa 3 3. 72 rssr
Practice: Evaluate. (solve)
1. 27 2. 43 3. 23 32 4. 42 25
Exponents 8.3
AlgebraOne of the easiest ways to multiply expressions using exponents is to writethem out in factored form, and then recombine terms using exponents:
Ex.32 62323)2(3 nnnnnnnnn
Practice: Simplify.
1. )2( 2xx 2. )(4 52 yxxy 3. 7121123 3)2( baba
Rules:When multiplying variables with exponents, simply add exponents:
Ex. 83553 nnnn or 5723522532 )( yxyxyxyx
Practice: Simplify.
1. )( 1225 xx 2. )7(3 14213015 baba 3. 203710 3)4( nmnm
The same rules apply for positive and negative exponents.
Practice: Simplify.
1. 43 35 xx 2. )3(4 4225 baba 3. 2832 yxyx
4. 2212 55 5. yxyx 11213 512
6. 3211 32 yy
Challenge: Find the Perimeter AND Area of each shaded figure below:note: all angles are right angles.
Exponents 8.3
ab5ab4
ab3ab7
x3x2x3
x5x3
AlgebraExponents and Division 8.5Review: Multiply.
1. )(3 524 yxx 2. 223 )4( yx 3. 523 )2(5 bbDividing Monomials:You can write-out variables and exponents, or simply subtract exponents:
Examples:
1. baba4
32
62
2. 29
115
412
yxyx
Practice: Divide/ Simplify. Answers should have positive exponents.
1. 4
33
2510
xyyx
2. ba
ba9
812
164
3. 26
33
1420
baba
What is a negative exponent?Look at the following pattern in our own number system:
876.543The 8 is in the ________ place .___102 The 7 is in the ________ place .___101 The 6 is in the ________ place .___100 The 5 is in the ________ place .___10 1
The 4 is in the ________ place .___10 2
The 3 is in the ________ place .___10 3
AlgebraExponents and Division 8.5Negative Exponents: A negative exponent can be expressed as a posi-tive exponent in the denominator:
Examples:
3
3 1x
x 4
4
1 aa
3
535
yxyx
5
2
2
5
yx
xy
Notice that a negative exponent in the denominator can also be ex-pressed as positive in the numerator.
Practice: Rewrite with positive exponents:
1. 2x 2. 3ab 3. 9
4
xy
4. 2
9
bx
5. 3)( ab 6.
baab
2
4
Negative Exponents: The easiest way to simplify expressions withnegative exponents is to begin by rewriting them:
Examples:
34
253baba
2
5
25
xyyx
Practice: Rewrite with positive exponents, then simplify:
1. 5
22
xx
2. 39
2
93
yxxy
3. 25
435baba
You can also use the subtraction method, but it becomes much moreconfusing.
AlgebraDivision With ExponentsName________________________ Period _____
Simplify each. Your answers should be written with positive exponents.
1. 7
2
xx
2. 3
2
abba
3. 2
5
412
yy
4. 5
33
217
axxa
5. xyx
2025 2
6. 1510
1030
126
baba
7. 4
3)(abab
8. 55
3
20)(50
babaab
9. 2
5
216
yy
10. 7
3
22
11.
baba2
35
42
12. 22
3
)6(2)3(aab
ab
2.3
AlgebraDivision With ExponentsName________________________ Period _____
More challenging problems:Simplify each. Your answers should be written with positive exponents.
13.
7
12
xx
14.
315
620
3521
baba
15. 4
5
)()(
abab
16.
2
22
3
yxyx
17.
3
5
2
2xx
18.
3
510
1030
baba
2.3
AlgebraRaising a power to a power:
Practice: Simplify each using what you know about exponents.
1. 42 )2( xy 2.
22 )2(3 baab 3. 5812 )2( yx
Examples: Raising a power to a power.
1. 325 )( yx 2.
6112 )2( ba 3. 2542 )(3 yxxy
Practice: Raising a power to a power.
1. 53 )(x 2.
22116 )3( ba 3. 552 )( baab
4.
5
3
3
yx
5. 211)( a 6.
322 )2( ba
Exponents 8.3
AlgebraExponents and Division 8.5Raising a fraction to a power: When raising a fraction to a power,apply the exponent to the numerator and the denominator:
Examples:
3
2
yx
3
5
222
y
yx 4
2
5
xy
Practice: Try these easy ones:Write-out if necessary.
1.
3
yx
2.
5
3
2
53
3.
4
3
42
yx
Practice: Try these more difficult problems.Like most of the math we have done, there are many ways to get theright answer. Answers should have positive exponents.
1.
5
5
2
xx
2.
2
39
23
yxxy
3.
5
2
4
42
xx
You can simplify what is in parenthesis before or after applyingthe exponent.
AlgebraExponents Reteach: MultiplyingName________________________ Period _____
Write each expression out without using exponents (write small!):
1. 435 yx 2.
32 )5( a 3. 233 )(5 xyx
Rewrite each expression using exponents:
4. babaaa 5. xxxxxx 333 6. 53 aaba
Simplify each using the rules for exponents.
7. )2(3 54312 baba 8. )2(3 145 xx 9.
272 )5(3 baab
10. 238 )5( ba 11.
5115 )2(5 xx 12. 2104 )3(2 baba
13. 2532 )6( xyyx
14. 543 )(2 xx 15.
2223 )2()( nnm
16. 753 3232 17. 235 )7(7 18.
2232 ])2[( ba
AlgebraExponents Reteach: DividingName________________________ Period _____
Write each expression out without using exponents (write small!): ex. xxxx 3
19. 7
3
62
xx
20. 3
32 )(aa
21. yxxy
2
2
10)2(
Simplify each, then rewrite each expression using exponents:
22. aaaaaaaa
10
523. bbaaa
baba
1555
24. yyyxxyxxx
Rewrite each with positive exponents: DO NOT SIMPLIFY, just rewrite using allpositive exponents:
25. 4x 26. 4
32
abba
27. 4
7
2
xyyx
28.
2
4
2
ba
Rewrite each then simplify: Take your time and complete several steps.
29. 142
308
497
baba
30. 5
23
6)3(
xyyx
31.
2
23
3
)(
mnnm
AlgebraQuick Review 1.7Cut-out the following and give each table a set (8 sets).Match the letters to the proper numbers to find the clue.ans: read it backwards (read backwards)
1. 222 )(abba
s. 4a
2. 222 )(abba d. 222 ba
3. 2
22
)(abba
r. 4
1b
4. 33312 )(abba
a. 615ba
5. 36
321
baba
w. 6
15
ba
6. 1117
52
baba
k. 15
6
ab
7. 1569462 5)( bababa c. 1564 ba
Algebra
8. 223129 )3( baba a. 8159 ba
9. 1181581511 2392 cbabac b. 8156 ba
10. 815815 25 baba t. 8156 ba
11. 44
4112baba
i. 8
152ba
12. 55
103
2
baba
d. 8
15
2ab
13. 156352 )2( baba a. 1567 ba
14. abababba 322 3 e. ababba 4322
15. babababa 66756 332 r. 753 ba
AlgebraPractice Quiz: Chapter 1 (4)Solve for a=3, b=5, c=2
1. acba 2
1.______
2. )(2 bacb 2.______
3. 2))(( abbc
3.______
Simplify:
4. 22 53 bb
4.__________________________
5. 41323 22 xx5.__________________________
6. 121115 44 aa6.__________________________
7. 101112 45 xxx
7.__________________________
8. baabbaab 2222 7243 8.__________________________
9. 3323 9559 aaaa
9.__________________________
Name________________________ Period _____
AlgebraPractice Quiz: Chapter 1 (4)Simplify: All answers should be written with positive exponents.
10. 35 xx
10.___________________
11. 33 92 abab
11.___________________
12. )2(5 26 baba12.___________________
13. 232 )2(4 xyyx
13.___________________
14. )6(7 38122 yxyx
14.___________________
15. 9
3
yy
15.______________
16. 5
3
aa
16.______________
17. yxyx
7
25
17.______________
18.
2
2
2
63
bab
18.______________
Name________________________ Period _____
AlgebraPractice Quiz: Chapter 1 (5/7)Solve for a=-3, b=5, c=2
1. acba 2
1.______
2. )(2 bacb 2.______
3. 2))(( acbc
3.______
Simplify:
4. 22 53 bb
4.__________________________
5. 41323 22 xx5.__________________________
6. 121115 44 aa6.__________________________
7. 101112 45 xxx
7.__________________________
8. baabbaab 2222 7243 8.__________________________
9. 3323 9559 aaaa
9.__________________________
Name________________________ Period _____
AlgebraPractice Quiz: Chapter 1 (5/7)Simplify: All answers should be written with positive exponents.
10. 252x10.___________________
11. 33 92 abab
11.___________________
12. )2(5 26 baba12.___________________
13. 238122 )3( yxyx
13.___________________
14. 22322 )2( yxyx14.___________________
15. 9
3
yy
15.______________
16. 3
8
1215
aa
16.______________
17.
2
77
23
412
baba
17.______________
18.
2
2
2
63
bab
18.______________
Name________________________ Period _____
AlgebraThe Distributive Property 1.7The Distributive Property states:
For any numbers a, b, and c:
acabcba )(
Examples:
distribute the 5 distribute the 3
155)3(5 xx 126)42(3 aa
srsr 1111)(11 22 232 )( xxxxx
Multiply the term outside the parenthesis by both terms inside.
Practice: Rewrite using the Distributive Property.
1. )2(5 yx 4. )1(9 2 x2. )2(7 yx 5. 2)134( b
3. )25(3 2 xx 6. )72(8 yy
Distributing the negative:
Ex:
distribute the -5 distribute the -a
)3(58 xx )42(7 2 aaadistribute the negative (-1).
)3(3 yyPractice: Rewrite using the Distributive Property.
1. )2(35 yxxxy 4. )3(45 2 a
2. )5(38 y 5. )4(6 yxx
3. )2(3 xx 6. )72()3( yy
AlgebraThe Distributive Property 1.7Practice: Rewrite the Following Using the Distributive Property:
1. )3(14 a 2. )52(4 25 cc
3. )52(2 aaa 4. 2)78( xx
5. )57( 22 xxyxy 6. 2)(157 abaaab
Practice: Fill-in the blanks. The GCF has been factored out for you.
1. 22 1814 abba ___)(___2 ab
2. yxyx 232 2115 ___)(___3 2 yx
3. aabab 3612 2 ___)___(___3 a
Practice: Factor the Following (Reverse the Distributive Property)
1. 23 65 aa 3. xxyx 3819 2
2. aba 1525 2 4. 12515 2 xx
AlgebraDistributing Division 1.7+You can use the Distributive Property with division.Example:
1. 6
1224x
Divide xx 46
24
and 2
612
Therefore 246
1224
xx
Practice:
1. 5
1510 x 2.
72142
x
3. xxx
63618 2
4. x
xx4
1612 2 5.
xxxx
22832 25
6. x
xx10
550 23
Practice: Answers will include fractions.
1. 20
1520 x 2.
xxx
362 25
3. x
xx4
38 4
Challenge Set: Some answers will include fractions.
1. x
xx7
146 34 2. x
xxx11
101122 23
3. 5
22
396
x
xx
AlgebraDistributive PropertyName________________________ Period _____
1.7+Rewrite and Simplify using the Distributive Property:
1. )5(7 xx 2. )52(4 a
3. )36(2 y 4. )52(8 2 aa
5. )72(2 xx 6. )2(2 yxxy
7. )34(6 aa 8. )5(32 xx
9. )6(46 aaa 10. )59(12 x
11. )(3 cabc 12. )73()24( 33 yxyx
AlgebraDistributive PropertyName________________________ Period _____
Simplify: Distributing division (‘bunny ears’):
13. 6
3612 x14. b
bab2
1220
15. 2
35
73549
bbb
16. yyyy
483216 23
17. bbbcab
5104015
18. 5
567
1282436
xxxx
19. aaaa
43128 23
20. 2
235
55410
xxxx
Factor (Rewrite using the Distributive Property in reverse):
21. ababba 1286 232 22. 2322 xyxyyx
1.7+
AlgebraDistributive Reteach: MultiplyingName________________________ Period _____
Rewrite each problem below using the Distributive Property.Multiply each term outside of the parenthesis by all terms inside the parenthesis.Careful with your signs and remember your rules for multiplying with exponents.
1. )(2 yx 2. )3(5 2 a 3. )2(5 yxx
4. )(2 yxx 5. )3(5 2 aaa 6. )2(5 232 xyxx
7. )(2 23 yyxxy 8. )3(5 223 abbaab 9. )2(5 3225 yxyx
Now try distributing some negatives. Remember your integer rules.
10. )(2 baa 11. )3(3 2 xx 12. )3(24 2 aa
For #12 above: DISTRIBUTE THE -2 NOT THE 4a2. Answer: 4a2-2a+6.Try the following similar problems and combine like terms wherever possible tosimplify your answer.
13. )2(35 23 baaa 14. )52( 3525 xxxx 15. )75(3 acc
16. )75(2 22 yxxyxy 17. )152(32 233 xxxx
AlgebraDistributive Reteach: DividingName________________________ Period _____
Rewrite each problem below using the Distributive Property.‘Bunny-Ear’ each term in the numerator with the term in the denominator.Careful with your signs!
18. 3
1215 x19.
xxx 23
20. x
xx5
1020 23
21. x
xyx5
5010 22 22. ax
axxa 232 23. xy
yxyx15
4530 223
24. 1
2
31521
x
x25. 22
233
yx
xyx26. 613
24211130
64812yx
yxyx
The final three answers involve fractions. The fractions should be simplified andleft as coefficients.
27. 3
122 x28. xy
yxyx20
2015 23 29. 2
23
10547
xxx
AlgebraReviewName________________________ Period _____
1.7+Rewrite and Simplify using the Distributive Property:
1. )(4 yx 2. )2(2 aa
3. )( 2 yxxy 4. )52(12 aa
5. )2(2 22 aabba 6. 3)1(5 a
7. )5(3 baab 8. )27(3 yxxxy
9. )2(5 23 aaa 10. cccbc )5( 23
AlgebraDistributive PropertyName________________________ Period _____
Simplify: Distributing division (‘bunny ears’):
11. a
aa2
84 2 12. xy
xyxy 37 2
13. 2
232
2144
babba
14. a
aaba 2
15. 2
3511
5152040
xxxx
16. m
mnmn28
1421 2
17. 3
3411
88616
cccc
18. ba
babba3
22 2
Factor Each: Reverse Distribution:
19. 23 3618 xyyx 20. bababa 2224 3
21. 3223 121640 xyyxyx 22. bababa 234 7545120
1.7+
AlgebraQuiz ReviewDistributive Property:
100. )3( aa 400. xx 3)7(5
200. )3(14 aa 500. )212(435 aa
300. )6(23 2 aaa 600. )( 223 yxxy
1.7+
Distributing Division:
100. a
aa2
146 2 400.
xxxx
43168 23
200. 2
24
7714
xxx
500. 4
32
3153
x
xx
300. a
aaa2
1262 34 600. yx
yxyx2
2233
2
Factoring:
100. xx 42 2 400. mnmnnm 345117 22
200. ababba 6189 22 500. 223 104143 yxyx
300. yxyx 222 1133 600. 345 13311991 xxx
AlgebraQuiz ReviewOrder of Operations:
100. )6)(5()2(15 400. 22 )52(4)3(
200. 2)4211(3 500. 22 )914()46(2
300. 3221 22
600. 2332 )2(2)2(
1.7+
Combining Like Terms
100. aa 113 400. 2222 272 yyxyyx
200. bbaba 355 3527 500. 22 )()3(2 xyxxy
300. baabba 333 533 600. )(4)(5 3242 yxxyyxx
Exponents:
100. 23 33 abab 400. 233 )3(2 aa
200. 45
7
yxxy
500. 232
333
)()(
baba
300. 5133 53 yxxy600.
232
12
)5(10
baab
AlgebraPractice Quiz 1.7+Solve for a=-3, b=5, c=2
1. )( accbc1.______
2. ))(( bcaac2.______
3. 23 ba3.______
4. cab 2
4.______Simplify:
5. 33 75 xyxy5.___________
6. baba 3656.___________
7. xyxy 111527.___________
8. 4343 53158 xyxxyx8.___________
9. 55 34 xx9.___________
10. 37 57 xx10.___________
11. 22)5(2 yy11.___________
12. )5(10 4511 abba12.___________
Name________________________ Period _____
AlgebraPractice QuizSimplify:
13. 9
3
yy
13.___________
14. 5
3
aa
14.___________
15. yxyx
7
25
15.___________
16.
3
7
2
63
bab
16.___________
Rewrite Using the Distributive Property and Simplify where possible:
17. )10(3 2a17._______________
18. )2(47 aa18._______________
19. )65(7 yxyxy19._______________
20. x
xx3
615 2
20._______________
21. a
aba6
6012 32
21._______________
22. x
xyx5
155 32
22._______________
Pledge: write-out and sign.
Name________________________ Period _____
1.7+
AlgebraFactoring 1.7+Reversing the Distributive Property is called Factoring.
Example: Rewrite the Following Using the Distributive Property:
1. )3( aa 2. )45(3 2 aa 3. )3(2 xxy
Answers should be:
1. aa 32 2. 23 1215 aa 3. xyyx 62 2
To factor an expression:a. Look for the GCF of all terms, including the variables.b. Place the GCF outside of the parenthesis.c. Divide each original term by the GCF to get the terms inside the
parenthesis.
Examples:
1. 22 1015 abba 2. xyyyx 6189 2
Practice: Fill-in the blanks.
1. ___)(___3912 2 ababba 2. )2___(147 22 baabba
3. ___)(___2 22 yyyx 4. )43___(86 235 yxxyx
Practice: Factor the following.
1. aba 156 2 2. xyyyx 18279 22 3. 22 73 xyx
Practice: Factor the following.
1. 22 642 babba 2. 684 3 aba
3. cxbxax 723654 4. 2352 72824 babba
AlgebraRewrite each by factoring (using the Distributive Property in reverse):
1. aa 244 2 2. 153 x
3. aa 1824 2 4. xx 1018 2
5. 23 cabc 6. 35 36 xx
7. 233 42162 ababba 8. aaba 562814 22
9. 22 1636 abba 10. xyyxyx 82210 423
11. 3223 648 bbaa 12. 3225 202515 xyyxyx
Factoring 1.7+Name________________________ Period _____
AlgebraWriting Expressions/ Equations 1.1Words to indicate:Addition Subtraction Multiplication Division
Rewrite as an expression:
1. The sum of seven and x.2. The quotient of a and b.3. Five times the sum of c squared and nine.4. Nine less than w.5. Twice a increased by nineteen.6. Two times the sum of a and nineteen.7. Half the product of x and y decreased by the quantity x plus 4.
In each case below, replace ‘a number’ with the variable n :
1. The sum of seven and a number.2. The quotient of a number and three.3. Five times the sum of a number squared and nine.4. Nine less than a number.5. Three times a number increased by ten.6. Two times the sum of a number and four.7. The product of a number and 3 increased by the number squared.
In the following problems, try to use variables that represent what isbeing given in the problem (For example, a could be used to repre-sent the number of apples. c could represent the cost. etc.)
1. The number of apples increased by six.2. Half the cost.3. Twice the number of cabs increased by three times the number of buses.4. Nine less than the number of days.5. Three times as many computers increased by ten.6. One third of the total number of boys and girls.7. The cost increased by 20%.
AlgebraWriting Expressions/ Equations 1.1In word problems, the word IS usually means equals.Terms with no equals sign are called expressions.If there is an equals sign, it is called an equation.
The following should be written as equations using variables.
1. The sum of seven and a number is 16.
2. Twenty is three times a number increased by ten.
3. One fifth of the total number of boys and girls is nine more than the num-ber of girls.
4. Four less than the number of pineapples is twice the number of pears.
5. Tom is three years younger than his sister Katie.
The word WHAT usually means USE A VARIABLE, often we use x.Ex. What is the sum of 2 and a number: x = 2 + n
1. What is the total number of cars and trucks?
2. What is 40% of the total cost?
3. What number is three times the sum of itself and seven?
Defining a variable:To solve many word problems, you must use a variable to represent an
unknown quantity (or quantities). Read the following example:
Margaret has a basket of apples and pears. The number of apples equalsthree more than twice the number of pears. If there are 15 pieces of fruitaltogether, how many apples and pears are there?
Using p for pears and a for apples, write two equations that could help yousolve this problem.
Write three equations:Amy is five inches taller than James. James is twice as tall as Pamela. Pamelais 41 inches shorter than Amy.
AlgebraWrite an expression for each statement below.If you need help, there is a list of answers on the back of the sheet to choose from. Write theexpression/equation AND the letter that goes with it. There will not be a word/phrase spelled.
1. Together, Alice and Betsy have $36.1.________________ _____
2. Nine less than a number.2.________________ _____
3. The product of nine and a number is four.3.________________ _____
4. Six times the difference of a and b is 36.4.________________ _____
5. Four less than the product of nine and a number isthe number itself.
5.________________ _____6. Nine less than the product of four and a number.
6.________________ _____7. Nine added to the quotient of a number and four.
7.________________ _____8. Nine decreased by a number.
8.________________ _____9. Four less than the product of nine and a number is nine
more than the product of four and the same number.9.________________ _____
10. Brenda has thirty-six less than Amy.10.________________ _____
11. Four less than the product of nine and a number.11.________________ _____
12. Nine more than a number.12.________________ _____
13. Nine times the sum of a number and four equals thesame number.
13.________________ _____14. Four times the sum of a number and nine.
14.________________ _____15. Three more than the quotient of two numbers.
15.________________ _____16. The product of two numbers is 36.
16.________________ _____17. One-fourth the sum of a number and nine.
17.________________ _____18. Nine is four more than Nancy’s age.
18.________________ _____19. Four more than the product of a number and nine.
19.________________ _____20. The product of nine and the sum of a number and four.
20.________________ _____
Name________________________ Period _____
Practice: Writing Expressions
AlgebraAnswer list for the front of the sheet: Each answer below appears once, one answer is unused.(note: There will not be a word or phrase spelled with the answers on the front.)
a. 9n b. 9n r. n9 d. 49n
w. )9(4 n y. 94 n
s. 36ab h. 49 n i. 9449 nn u. ba
3k. nn )4(9
m. 36 ba n. 36 ab p. 3649 n q. 36)(6 ba
c. )4(9 n t. 49 n o. 49 n e. 49 n
x. 49 n f. nn 49
Simplify each expression below and find the answer above. The letters will create a phrase.
21. 2377 nn _______ 22. 2
23
2818
mnmnmn
_______ 23. 2
223ab
baab _______
24. )7(3)3(4 nn _______ 25.
n
3133
_______ 26. nnn
436 2
_______
27. 3689 abab _______ 28. nn 12331 _______ 29. nnn
5455 2
_______
30. 8272 n
_______ 31. )5.1(4)3(5 nn _______
32. )9(44
412
313
aababa
_______ 33.
31612
43 n
_______
answer:
_____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ ! 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Name________________________ Period _____
Practice: Writing Expressions
AlgebraQuick Review 1.7This set of equations can be grouped into sets of letters whichcan be rearranged into words that form a question.To group the letters, find expressions or equations that areequal. Rearrange the letters in each set to form words and rear-range the words to form a question.Raise your hand when you know the answer to the question.
W. )3( yxx H. xyxxyx 223 22
A. 3
45 3x
yxx T. )124(
41 2 xyx
I. x
xyx2
4 2 S. xyyx
61
32
T. )5(2 x H. 2
23
4408
xxx
E. Twice a number decreasedby ten.
C. 333 xxyyxxy U. 2
22
x
yxx
B. )2( 12 yxxx
E. Twice the cube of a numberdecreased by y.
Algebra
R.
3
13
5
48
yx
xO. 6
37
324
xyyx
O. 3636 113 yxyx T. )32_____(
241643
7639
yxyxyx
O. )(2 yxx F. xyxxxy 6534 22
E. xyxyyx 23
I. )42(21 11911 xxx
G. 2
42 2 xH. (_______)5
1052
223
xyxyyx
T. 22 8375 xx
?. The sum of a numbersquared and two.
AlgebraTest ReviewLike Terms:
100. xx 82 400. axaxax 325
200. 22 3aa 500. )( yxyx
300. 232 53 xxx 600. xaaxa 222 5)5(
1.7+
Factoring:
100. xyx 22 2 400. 3432 2410 ababba
200. 227 217 yxx 500. 234 304515 xxx
300. 22 5642 xyyx 600. aa 187143 2
Distribution:
100. 2422 yx
400. )25(3 22 xx
200. )6(2 23 xxx 500. 3
32
3915
x
xx
300. xyyxyx
31218 3223
600. )53( 4523 xyxyx
AlgebraTest ReviewExponents:
100. )3(2 23 yxxy 400. 21230 )5(2 aa
200. 5133 53 yxxy500. 232
12
)5(10
baab
300.
22
546
xx
600. 3232 ])[( ba
1.7+
Equations and Expressions: Write.
100. Twice the sum of a number and seven.
200. The quotient of four and a number is increased by 12.
300. Mary is five years older than seven times her dog Peaches’ age.
400. The cost of a cab ride if you pay $3.50 per mile, and tip thedriver $5 is $90.
500. A rectangle is twice as wide as it is long. The area of the same rectangleis 52 inches squared. Write two equations to describe this rectangle.
600. In a right triangle, the shortest side is five inches less than half as long asthe longest side. The middle side is four inches longer than twice theshortest side. The longest side is three times the shortest side. Draw thetriangle and label all three sides, using x for the shortest side.
Algebra
1. Simplify 2222xxxx 2. Simplify
1
2
2
1
xxxx
3. Distribute 111 xxx 4. Simplify
x
xxx6
666 23
5. Distribute )))((( babababa 6. Simplify
43210
yx
yx
yx
7. Greg, Hank, Iris, Josh, and Kelly each collect silver dollars. Kelly has three more than twice as many asJosh, who has three more than twice as many as Iris, who has three more than twice as many as Hank, whohas three more than twice as many as Greg, who has three. How many does Kelly have?
8. Combine like terms:
yxyxyxyxyxyx 101099....443322
9. Factor the answer above.
Name________________________ Period _____
Tricky Review Problems
Algebra10. Factor
434445 xxx
11. Which is greater, 432 or 234 ?
12. Solve for x: x9819
13. Find the numerator: xyzxyxzyz
432
Challenge: When you solve the following, how many zeroes are in the answer? 9111010 1010
Name________________________ Period _____
Tricky Review Problems
AlgebraSimplify:
1. 22 57 xxx
1.__________________________
2. 22222 72 xyyxyxyx
2.__________________________
3. 323 592 bbbbb
3.__________________________
4. )9(5 33 yxyx4.__________________________
5. )7(4 22 baba5.__________________________
6. 22225 )5(4 yxyx
6.__________________________
7. yxyx
4
3
105
7.__________________________
8. 35
32
212
baba
8.__________________________
9. 2
222
16)2(
xyyx
9.__________________________
Name________________________ Period _____
1.9Practice Test (5,7)
AlgebraWrite an expression or equation for each:
10. Five times the difference of a number and eleven.10.__________________________
11. Claudia is eleven years older than her brother James.11.__________________________
12. The quotient of x and five decreased by twelve is fifteen.12.__________________________
Rewrite each using the Distributive Property:
13. )6(53 2 a13.__________________________
14. )2(37 xx 14.__________________________
15. )78(4 abab15.__________________________
16. aaa
51015 2
16.__________________________
17. xyxyyx
31512 33
17.__________________________
18. xyyxyx
329 234
18.__________________________Factor Completely (reverse distribution):
19. babab 21714 2 19.__________________________
20. 223 2821 yxyx
20.__________________________
Name________________________ Period _____
1.9Practice Test (5,7)
AlgebraPractice Test (4th) 1.7+Solve for a=4, b=-5, c=2
1. )( accbc1.______
2. ))(( bcaac2.______
3. 23 ba3.______
Simplify:
4. 22 7 aa 4.___________
5. xyxxyx 53 22 5.___________
6. 332 9547 cccc 6.___________
7. )9(7 22 cc 7.___________
8. )9(2 4592 yxyx8.___________
9. 322 )2(10 aba9.___________
10. 5
15
aa
10.___________
11.
2
2
24
xyyx
11.___________
Name________________________ Period _____
AlgebraPractice Test (4th)Rewrite Using the Distributive Property and Simplify where possible:
12. )3(2 xx12._______________
13. abbaa )3(313._______________
14. )4(3 yxxy14._______________
15. xxx
51530 2
15._______________
16. aacab
72114
16._______________
17. xyxyyx
21232 25
17._______________
Write each sentence as an algebraic expression or equation. DO NOT TRY TO SOLVE OR SIMPLIFY.
18. Meredith is three years older than her cousin Nina.18.____________________
19. Three less than twice the square of a number.19.____________________
20. Six more than the number of cars.20.____________________
21. Four times the sum of a number and two is eight less than thesame number.
21.____________________
22. The quotient of x and y is three less than the product of x and y.22.____________________
Name________________________ Period _____
1.7+
AlgebraPractice Test 2 (4th)Solve for a=4, b=-3, c=2
1. 22 abba 1.______
2. bcab )(2.______
3. ccaab 2)(
3.______
Simplify:
4. ababba 97 22 4.___________
5. xyxxyx 933 5.___________
6. 5252 627 xxxx 6.___________
7. )2(9 2225 cbcb 7.___________
8. 345 )2( ba8.___________
9. )12(10 22 abab9.___________
10. 5
5
xx
10.___________
11.
3
3
24
xy
yx
11.___________
Name________________________ Period _____
AlgebraPractice Test 2 (4th)Rewrite Using the Distributive Property and Simplify where possible:
12. )3( 32 aaa 12._______________
13. 2)(5 xyxx 13._______________
14. )3(53 xxx14._______________
15. xxx
61824 23
15._______________
16. babb
72114 2
16._______________
17. xyxyyx
6342 3
17._______________
Write each sentence as an algebraic expression or equation. DO NOT TRY TO SOLVE OR SIMPLIFY.
18. Tim has 19 dollars more than Rachel.18.____________________
19. The sum of a number squared and ten is twenty-six.19.____________________
20. Five less than x is divided by three.20.____________________
21. Four more than the quotient of a number and two.21.____________________
22. The difference of x and y is nine more than x cubed.22.____________________
Name________________________ Period _____
AlgebraAn equation is like a balance. Each side is equal to the other.In order for an equation to remain balanced, what you do to one sideyou must also do to the other.
Animals (seals, aardvarks, woodchucks and a couple elephants) are play-ing on the see-saws at the playground. Wow!
You notice the following animals balanced perfectly on each of the see-saws (forget about see-saw physics and animal behavior for a moment).
Group 1Left side: Right side:1 Elephant, 6 seals 23 seals
1. How many seals equal one Elephant?
Group 2Left side: Right side:4 Aardvarks, 7 Woodchucks 31 Woodchucks
2. How many woodchucks equal one Aardvark?
Group 3Left side: Right side:13 Seals, 20 woodchucks 1 Elephant
3. How many woodchucks equal the weight of one seal?
4. Which weighs more - a seal or an aardvark?
Practice: Solve the following one-step equations.
1. 142 x 2. 53 a
3. 93
x
4. 1253
x
2.1Intro to Solving Equations
AlgebraSome Equations require more than one step to solve.When solving Multi-Step Equations, work in Reverse Order of Op-erations to solve for the variable. This means you must undo additionand subtraction, then multiplication and division followed by what is left inparenthesis (or other grouped operations).
We will deal with exponents later.
Examples:
1. 1842 x 2. 1935 a 3. 27
24
b
Practice:
1. 753 x 2. 537 a 3. 38
73
b
More Difficult Practice:
1. 1543
x 2. 12)3(2 y 3. 724
3
a
Writing and solving equations:Define a variable, write an equation, and solve it:
Example:Martin is seven years older than half Maria’s age. If Martin is 15, how oldis Maria?
Practice:1. Phillip has three more than twice as many teeth as his grandpa. IfPhillip has 27 teeth, how many does grandpa have?
2. In 2007, Joey Chestnut ate 12 less than twice as many hot dogs asSonya Thomas in the Nathans Famous hot dog eating contest. Joey ate66, how many did Sonya eat?
2.2Solving Multi-Step Equations
AlgebraWhen solving an Equation with the Variable on Both Sides, youmust move the variable so that it is only on one side of the equation.
Examples:
1. 19582 xx 2. aa 935
Practice:
1. 85 xx 2. 9243 yy
3. 1152 xx 4. xx 62
33
Some equations require some work before you can move the variable.
More Difficult Examples:
1. xx 15)62(3 2. aa
62
5
More Difficult Practice:
1. 1)45(2 xx 2. xx 72
21
3. 175
3 xx
4. 17
32
xx
2.4Variable on Both Sides
AlgebraTwo Steps Practice. Solve for x.
1. 3253 x 4. 746
x
2. 181115 x 5. 31520
x
3. x32418 6. 161043
x
Both Sides Practice: Solve for x.
1. 175 xx 2. xx543
21
3. xx
5
1234. 8)3(2 xxx
Two Steps Word Problem Practice.Define a variable, write an equation and solve:
1. A rental company rents big-screen televisions for $22 down plus $4 aday. If the final bill is $46, how many days did you rent the televisionfor?
2. At a factory, you can make 14 widgets per hour. If you already havefifteen widgets made, how many hours will it take you get up to 71widgets?
3. You are buying a pair of jeans and several shirts at the store. If thejeans cost $40, and the shirts are $11 each, how many shirts canyou buy if the total is $106?
2.4Solving Equations
AlgebraMost students hate fractions.Good news! They are easy to get rid of before you begin solving anequation!
Examples:
1. 41
125
21
32
xx 2. 232
51
xx
Practice:
1. 81
43
21
41
xx 2. 61
322
92
xx
The same can be done with decimals!This is not as common or necessary, but decimals can be removed as wellwith some simple multiplication.
Examples:
1. 85.25.435.41.2 xx 2. 16.29.254.02 xx
Practice:
1. 7.21.25.15.3 xx 2. 44.008.003.055.0 xx
2.2Removing Fractions & Decimals
AlgebraEquations Review (5/7)Name________________________ Period _____
2.4Easier Practice.Solve for x. Check your work. Simplify Fractional Answers. NO CALCULATOR. SHOW ALL WORK.
1. 472
x 2. xx 1.48.32.2
3. xx 723 4. 8.42.18.2 xx
5. 8432
x
6. 6.41.13.1 x
7. 141
43
43
xx 8. xx 52
3
9. 9)25(3 x 10. )1(417 xx
AlgebraEquations Review (4)Name________________________ Period _____
2.4Practice.Solve for x. Check your work. Simplify Fractional Answers. NO CALCULATOR. SHOW ALL WORK.
11. 6352
x 12. 5.41.32.0 x
13. xx
3
5214. 75.1535.11.2 x
15. xx 283
2 16. 9.17.53.21.1 xx
17. 723
1
xx18.
1213
32
21
xx
19. 4528
21
xx 20. )1(42
3
xx
AlgebraEquations Review (5/7)Name________________________ Period _____
2.4Practice.Solve for x.
11. xx 283
2 12. 9.17.53.21.1 xx
13. 723
1
xx14.
1213
32
21
xx
15. 4528
21
xx 16. )1(42
3
xx
17. wa
yx
18. dcxa )(
19. fg
dcx
20. zybax )(
AlgebraEquations ReviewName________________________ Period _____
2.4Solve for x:
1. 4212 xx 2. 256 xx
3. xx 5)9(2 4. xx 57
5. 9432 xx 6. xx 2)8(2
7. 3314
32
xx 8. xx 52
153
9. )1(3)7(5 xx 10. 43
2
xx
11. 21
327
41
xx 12. aa513
32
92
AlgebraEquations ReviewName________________________ Period _____
2.4Write an equation and solve:
13. Five more than twice a number is 11. What is the number?
14. Twice the sum of a number and six is equal to the product of the number and three.
15. Three more than four times a number is equal to nine less than twice the samenumber?
16. Four times the greater of two consecutive integers is equal to three times the lesser.Find the two integers.Hint: If two numbers are consecutive and the first is ‘x’, then the second would be ‘x+1’.
AlgebraSolving Formulas 2.6By now, you should be able to recognize the steps to take to-ward solving simple equations.
Examples:What steps (in the correct order) would you take to solve for x?
1. 915
82
x2. 387)1(4 x
a. Multiply both sides by _______. a. _______________________b. Subtract _____. b. _______________________c. Divide by _____. c. _______________________
You can apply similar steps to solve formulas.
Examples:What steps (in the correct order) would you take to solve for x?
1. cb
yax
2. ybcxa )(
a. Multiply both sides by _______. a. _______________________b. Subtract _____. b. _______________________c. Divide by _____. c. _______________________
Practice: Solve for x.
1. cbax 2. fcax
3. vw
xa
4. ya
cdxb
AlgebraSolving FormulasName________________________ Period _____
Solve for x:
1. ac
yx
2. dcxa )(
3. ady
bx 4. ycx
ba
)(
5. dycxa
6. cdcax
7. abxyx 2 Challenge: d
gabax
AlgebraPractice:Determine whether each pair of fractions are equal by making their de-nominators equal. Fill the blank with a >, < or =.
1. 75__
54
2. 97__
76
3. 96__
128
If two fractions are congruent,their cross products will always be equal.
Practice:In the following proportions, use cross-products to solve for the variable.Simplify fractional answers.
1. 52
15
x2.
43
32
x
3. 31
41
x
Example: Solve for x.
1. 13
59
2
xx
Practice: Solve for x.
1. 49
1 x 2.
23
75 x 3.
54
72
xx
More Difficult Practice. Solve for x. Simplify fractional answers.
1. 1
5.12.2
xx
2. 14
1027
5
xx
3. 13
4
x
x4.
312
47
x
Solving Proportions 2.6+
AlgebraProportion Equations ReviewName________________________ Period _____
2.6+Solve for x. Show all your work. Use Cross-Products and simplify fractional answers.
1. 159
3 x 2.
129
83
x
3. 315
x 4.
145
710 x
5. x10
95 6.
512
103
x
7. 91
155
x
8. 211
1216
x
9. 5
135
x10. 2
1014
x
11. 30
3352
x12.
xx 152
413
AlgebraProportion Equations ReviewName________________________ Period _____
2.6+Write an equation, then solve. Show all your work.
1. The quotient of a number and 28 is ¾. What is the number?
2. Thirty divided by 4 is equal to the quotient of a number and eight. What is the number?
3. 24 divided by a number is equal to two-thirds. What is the number?
4. The product of 31 and x is equal to the sum of x and 150. Solve for x.
AlgebraUse Inverse Operations to solve equations.Undo Addition with Subtraction.Undo Multiplication with Division.Undo a Square Root by Squaring.
Examples: Solve for x.
11x 51 x 106 x
Before you can square both sides, you must Isolate the Radical.The radical sign is another grouping symbol. Undo everything else beforesquaring both sides.
More Examples: Solve for x.
1. 102x 2. 167 x
3. 7511 x 4. 4535 x
Practice: Solve for x.
1. 1675 x 2. 1152 x
3. 2531
x 4. 114185 x
Challenge: Solve for x.
1. 25
23
x2. 2
692
x
Solving Square Root Equations 2.7+
AlgebraSquare Root Equations ReviewName________________________ Period _____
2.7+Solve: Isolate the radical. Square Both Sides. Solve.
1. 12x 2. 52 x
3. 97 x 4. 1024 x
5. 1232 x 6. 1262 x
7. 8932
x 8. 539
x
9. 13716 x 10. 437205 a
11. 4237 x 12. 8352
x
13. 52
154
x
14. 63
84
x
AlgebraSquare Root Equations ReviewName________________________ Period _____
2.7+Write an equation and solve:
15. Five more than the square root of a number is 11. What is the number?
16. Five times the greater of two consecutive integers is equal to 21 more than thelesser. Find the two integers.Hint: If two numbers are consecutive and the first is ‘x’, then the second would be ‘x+1’.
17. The shortest side of a triangle is 7 inches shorter than the longest side. The middleside is twice as long as the short side. Find the length of all three sides if theperimeter is 31 inches.
18. Four times the square root of a number is 18. What is the number?
AlgebraExamples: Solve for x.
5x There are two solutions: 5 and -5
Name the two solutions to each equation (or write NO SOLUTION):
11x 13x 3x 62 x
As in square root equations, you must Isolate the Absolute Valuebefore giving two solutions:
More Examples: Solve for x.Name the solutions to each equation:
115 x 303 x 195 x
Sometimes you need to finish AFTER setting up two equations:More Examples: Solve for x.Name the solutions to each equation:
1153 x 3023 x 192
15 x
Practice: Solve for x.
1. 1532 x 2. 2423 x 3. 2375115
x
Absolute Value Equations 2.7+
AlgebraAbsolute Value EquationsName________________________ Period _____
2.7+Solve: Each problem (#1-12) will have two solutions.
1. 12x 2. 52 x
3. 97 x 4. 1532 x
5. 2012 x 6. 12242 x
7. 8552
x 8. 123
9
x
9. 1372 x 10. 35535 x
11. 92
224
x 12.
53
25
x
Challenge. xx 23
AlgebraBasic Equations:
Practice. Solve for x.
100. 139 xx 200. cyxda
300. 5322
21
xx 500. 411
322
xx
Square Roots Equations:
Practice. Solve for x.
100. 1412 x 200. 33
52
x
300. 53
52
x500. dbcxa
Absolute Value Equations:
Practice. Solve for x.
100. 122 x 200. 12623 x
300. 12
52
x500. 39
32
512 x
Quiz Review 2.7+
AlgebraPractice Quiz : ConceptsName________________________ Period _____
Solve for x: SIMPLIFY all fractional answers. WRITE INFINITE OR NO SOLUTIONSwhere applicable.
1. 32
1
x
1. x=_______________
2. xx 7532. x=_______________
3. 513
32 xx
3. x=_______________
4. 43
52
xx
4. x=_______________
5. 11217 x5. x=_______________
6. )93(2)62(3 xx6. x=_______________
7. 23
72
x7. x=_______________
8. xx 3)12(3 8. x=_______________
2.7+
AlgebraPractice Quiz : ConceptsName________________________ Period _____
2.7+Solve for x: SIMPLIFY all fractional answers. WRITE INFINITE OR NO SOLUTIONS
where applicable.
9. yb
cx
9. x=_______________
10. caxy 10. x=_______________
11. 539 x11. x=_______________
12. 12282 x12. x=_______________
13. 953 x13. x=_______or________
14. 832 x14. x=_______or________
15. 832
3
x
15. x=_______or________
AlgebraSolve Inequalities just as you would Equations.There is ONE DIFFERENCE you must remember:Whenever you MULTIPLY or DIVIDE both sides by a negative,REVERSE THE DIRECTION OF THE > or <.
Examples:
1. 226 x 2. 53
11
x3. 9
43
x
_____________________________________________________________-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Practice:
1. 104 x 2. 73
15
x3. 6)312(2 x
_____________________________________________________________-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Practice: Fractions.
1. x214 2.
31
652 x 3. x
32
214
_____________________________________________________________-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
Challenge: Proportions. 95
312
x
x Check your answer <, >.
Solving Simple Inequalities 3.4
AlgebraCompound Inequalities:
Ex.
6315 x In this compound inequality, 3x is less than 15 butgreater than -6. Solve each separately.
x315 63 x
The answer is written: 25 xAND/OR Compound Inequalities.Graph each of the following on a number line (separately).
1. 25 x 2. 25 x
_____________________________________________________________-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
The word AND is used to show where two graphs overlap (where bothparts are true).The word OR is generally used when there is no overlap (two parts can’tboth be true).
Examples: Solve and graph.
1. 51329 x 2. 22 xx or 732 x
Practice:
1. 5327 x 2. 222 xx
3. x26 or 75 x 4. 83
4
x or 1)1(2 xx
Solving Compound Inequalities 3.5
AlgebraAbsolute Value InequalitiesLike equations, absolute value inequalities have two solutions.
Ex.
8x Think of some possible values for x that make this true.
x is either ______ than 8 or ____ than ____.
3x Think of some possible values for x that make this true.
Write the solution set: ____________
Ex.
63 x In this inequality, 3x could be greater than 6 OR
3x could be less than -6. (If you don’t see why, ASK!)
Setup two inequalities. 63 x 63 xWrite the answer as a compound inequality.
ISOLATE THE ABSOLUTE VALUE FIRST then setup 2 inequalities.Examples:
1. 30218 x 2. 23
42
x
Practice:
1. 9323 x 2. 2523 x
3. 1126 x 4. 1733
42 x
Absolute Value Inequalities 3.6
AlgebraSimple Inequalities:
Practice. Solve. Sketch a graph for each.
100. 428 xx 200. 4329 x
300. 51
322
21
xx 500. 41
622
xx
Absolute Value Inequalities:
Practice. Solve. Write answers using AND or OR.
100. 93 x 200. 632
x
300. 42
6
x500.
311
212
x
Compound Inequalities:
Practice. Solve. Sketch a graph for each.
100. 122 x or 462 x 200. 10624 x
300. 1224 xx 500. xx 241
322
Solving Inequalities Review 3.6+
AlgebraPractice Quiz (4)Name________________________ Period _____
3.6Solve for x: Solve the following EQUATIONS.
1. 32
533
x
x
1. x=_______________
2. 252 x2. x=_______________
3. 12515 x3. x=_______________
4. 472 x4. x=_______________
5. 4823 x5. x=_______or________
6. 832 x6. x=_______or________
7. 104
)2(5
x
7. x=_________________
AlgebraPractice Quiz (4)Name________________________ Period _____
3.6Solve for x: Solve the following INEQUALITIES.WRITE AND GRAPH YOUR ANSWER. ex: x>-2
CIRCLE AND or OR for #11-14
8. 52
3
x
8. ___________ ____________________
9. 573 x9. ___________ ____________________
10. 253 x
10. ___________ ____________________
Use a separate sheet for work:
11. 462 xx11. _______ and/or _______ ____________________
12. 52
32
xx
12. _______ and/or _______ ____________________
13. 73 x or xx 12513. _______ and/or _______ ____________________
14. 2)1(2 xx14. _______ and/or _______ ____________________
-3 -2 -1
AlgebraPractice Quiz (5,7)Name________________________ Period _____
3.6Solve for x: Solve the following EQUATIONS.
1. 32
533
x
x
1. x=_______________
2. dbax 2. x=_______________
3. 12515 x3. x=_______________
4. 472 x4. x=_______________
5. 4823 x5. x=_______or________
6. 83
32
x
6. x=_______or________
7. ad
cbx
)(
7. x=_________________
AlgebraPractice Quiz (5,7)Name________________________ Period _____
3.6Solve for x: Solve the following INEQUALITIES.WRITE AND GRAPH YOUR ANSWER. ex: x>-2
CIRCLE AND or OR for #11-14
8. 52
3
x
8. ___________ ____________________
9. 573 x9. ___________ ____________________
10. 253 x
10. ___________ ____________________
Use a separate sheet for work:
11. 35 x11. _______ and/or _______ ____________________
12. 52
32
xx
12. _______ and/or _______ ____________________
13. 73 x or xx 12513. _______ and/or _______ ____________________
14. 21012 x14. _______ and/or _______ ____________________
-3 -2 -1
AlgebraGeometric Word ProblemsTriangles and their angles.Use to solve the following: The sum of the angles in a triangle will alwaysequal 180 degrees.
Ex: The measure of the smallest angle in a triangle is half the measureof the largest angle. The largest angle is 15 degrees greater than themedium angle. List the measures of all three angles.
Practice:1. A triangle’s smallest angle is 55 degrees smaller than its largest angle.The middle angle is 5 degrees larger than the smallest angle. What arethe measures of all three angles?
2. Of the two smaller angles in a right triangle, one measures twice aslarge as the other. What are the three angle measures?
3. The smallest angle in a triangle is only one-third as large as the middleangle. The largest angle is eight times the sum of the two smaller angles.Find all three measures.
Area and Perimeter Problems.Perimeter is the distance around a figure (add all sides).Area of a rectangle equals base times height (length times width).
Ex. The length of a rectangle is five inches greater than its width. Theperimeter of the rectangle is 38 inches. What is the area of the samerectangle?
Practice:1. What is the area of a rectangle whose length is three inches morethan twice its width if the perimeter is 36 inches?
2. What is the area of a rectangle if the width is five centimeters lessthan half the length, and the perimeter is 32 centimeters?
3. In a scalene triangle, the short side is half the longest side and themiddle side is two inches longer than the shortest side. Find the lengthsof all three sides if the perimeter is 22 inches.
Challenge: An equilateral triangle and a rectangle have the same pe-rimeter. The width of the rectangle is three inches longer than the sidesof the triangle. If the rectangle is 8 inches tall, what is its area?
2.5+
AlgebraGeometry Word ProblemsName________________________ Period _____
2.5+Sketch, write an equation, and solve:1. The smallest angle in a triangle is ten degrees smaller than the medium angle, andthe largest angle is ten degrees more than twice the medium angle. What are the anglemeasures in the triangle?
equation:_________________
solutions:_______ _______ _______
2. Two congruent angles in an isosceles triangle are nine degrees smaller than thelarger angle. What is the measure of the two congruent angles?
equation:_________________
solution:_______
3. In a pentagon, the sum of the angles is 540o. If all five angles are congruent, whatis the measure of each angle?
equation:_________________
solution: _______
4. The largest angle in an obtuse scalene triangle is three degrees greater than threetimes the smallest angle. If the medium angle is 57o, what are the measures of theother two angles?
equation:_________________
solutions: _______ _______
5. The largest angle in a pentagon is 4 degrees greater than the next largest, which is3 degrees greater than the next largest, which is 2 degrees larger than the next, whichis one degree larger than the smallest angle. What are the five angles? (See #3 formore info.)
equation:________________________________________
solutions: _______ _______ _______ _______ _______
AlgebraGeometry Word ProblemsName________________________ Period _____
2.5+Sketch, write an equation, and solve:6. A rectangle’s length is 19 inches greater than its width. If the perimeter is 50inches, what is the area of the rectangle?
equation:_________________
solution:_______
7. The perimeter of a rectangle is 52cm. If the width is 1cm less than twice the length,what is the length of the short side of the rectangle?
equation:_________________
solution:_______
8. The perimeter of a pentagon is 58 inches. If three of its sides measure 9 inches,and one of the remaining two sides is five inches longer than the other, what is thelength of its longest side?
equation:_________________
solution:_______
9. The height of a rectangle is nine inches less than its perimeter and the width of thesame rectangle is 6 inches less than its perimeter. What is the area af the rectangle?
equation:_________________
solution:_______
10. A square and a rectangle share the same perimeter. If the height of the rectangleis twice the height of the square, and the length of the rectangle is three inches lessthan half the length of the square, what is the area of the square?
equation:_________________
solution:_______
AlgebraWord ProblemsNumbers and ‘Consecutive Integer’ Problems:Define a variable and solve:
Ex: The sum of two consecutive integers is 79.What are the two integers?
Ex: The sum of two integers is 44. The difference of the sameintegers is 18. What are the integers?
Practice:1. The sum of two consecutive integers is 51.
What are the integers?
2. The sum of two consecutive odd integers is 24.What is the product of the same two integers?
3. I am thinking of three numbers whose sum is 31. The first numberis three less than the second and the third number is twice the first.What are the three numbers?
Equal Amounts:You can write an equation comparing two values that are equal.
Ex: John and Marge both paid the same amount for their taxi rides, butJohn went 2 miles farther than Marge. John’s taxi charged a $2 fee plus$.50 per mile, and Marge’s taxi charged a $5 fee plus $.25 a mile. Howfar did Marge ride in her taxi?
Ex: Nora flies her plane from NY to Chicago into the wind at 160mph,then flies back with a tailwind at 200mph. If it took an hour longer to gothan to come back, how far apart are NY and Chicago? (hint: d=rt)
Practice: THINK! Set-up two equations that are equal.
1. Julia can run 2mph faster than Nikhita. Nikhita takes 2 hours to finishthe race, while Julia finishes in 90 minutes. (hint: Units matter.)a. How fast does Julia run? b. How far did they run?
2. Mark and Robert are both driving to Orlando. Mark leaves at 9am andaverages 60mph, while Robert leaves an hour later but drives 70mph.a. At what time does Robert pass Mark on the highway.b. How many miles have each driven when Robert passes Mark?
2.5+
AlgebraTwo Numbers Word ProblemsName________________________ Period _____
2.5+Write an equation and solve:1. The sum of two numbers is 9 and their difference is 7.
What are the two numbers?
equation:________________
solutions:_______ _______
2. The sum of two consecutive integers is 55. What is the smaller of the two num-bers?
equation:________________
solution:_______
3. One number is equal to six more than five times another number. If the sum of thetwo numbers is twelve, what it the product of the two numbers?
equation:________________
solution:_______
4. The sum of three consecutive odd integers is -3. What are the integers?
equation:________________
solutions: _______ _______ _______
5. Challenge. I am thinking of two numbers. The first number is 19 less than threetimes the second. The second number is 8 more than twice the first. What are the twonumbers?
equations:________________
________________
solutions: _______ _______
AlgebraEqual Amounts Word ProblemsName________________________ Period _____
2.5+Write an equation and solve:6. Kelly and Kate each spent the same amount buying fruit. Kelly bought five pearsand an apple, while Kate bought two pears and three apples. If apples cost $0.45, howmuch do pears cost?
equation:_____________________
solution: _______
7. Hannah and Rachel scored the same number of points on a test. Hannah got 5 ofthe 10-point questions right and 3 of the other questions correct. Rachel got just oneof the 10-point questions right, but 11 of the other questions correct. How much werethe other questions worth?
equation:________________
solution:_______
For the last three problems, use the formule d=rt (distance equals rate x time).Rate means ‘speed’. In each problem, find two distance equations that are equal.8. Deepthi and Austin were in a race. Deepthi gave Austin a 3-second head start, andthey crossed the finish line at the same time. If Deepthi runs 8 meters per second andAustin runs 7 meters per second, how many meters long was the race?
equation:________________
solution:_______
9. Demetri rode his bike to Caleb’s house. On the way there, he went up a lot of hillsand averaged only 12 miles per hour. On his way home it was mostly downhill and heaveraged 18 miles per hour. If it took a half hour longer to go than to come back, howfar is it from Demetri’s house to Caleb’s house?
equation:________________
solution: _______
10. Allison and Mason each leave Ligon at the same time to walk to El Rodeo restau-rant. Allison walks 3 miles an hour and arrives 15 minutes before Mason, who walked2 miles an hour. How far away is El Rodeo?hint: distance = speed(time) ... d=rt
equation:________________
solution:_______
AlgebraWord ProblemsWord problems: Inequalities.Simple:Ex: The Ligon Student Council is organizing a fundraiser to raise moneyfor Cancer research. There are 58 homerooms, and they have alreadyreceived a contribution of $1,000 from the PTA. How much money doeseach homeroom have to collect in order to guarantee a total of at least$8,250?
Compound:Ex. Robert has $5 more than twice as much in his wallet as David has. IfRobert has between $7 and $15, how much does David have?
Practice: Write an inequality for the following.1. In order to make the state track finals, Connor has to beat his timefrom last year by at least 6 seconds. Last year he ran the quarter-mile in67 seconds. Write and solve a simple inequality to represent the time heneeds to run this year.
2. The width of a rectangle is three inches greater than half the length.Write a compound inequality to represent the length if the width is be-tween 8 and 11 inches.
Challenge: Write a compound inequality to represent both the area (a)and perimeter (p) of the rectangle described above.
Real-life absolute value inequalities.
Ex: Write an equation for the following: Mapquest states that the drivefrom here to Orlando will take 11½ hours, give or take an hour. Write aninequality for this using absolute value.This is saying that the POSITIVE difference between your drivetime and 11.5 hrs is < 1 hr.
Practice: Write an absolute value inequality for the following.1. Tomorrow’s high temperature will be within three degrees of 68o.
2. The governor’s favor ability rating is 57%. The margin of error in thepoll is +/- 3%.
Challenge: Between 60% and 80% of Americans stink at writing abso-lute value inequalities. (Hint: Think of this as a margin of error problemlike the one above.)
2.5+
AlgebraInequality Word ProblemsName________________________ Period _____
2.5+Simple InequalitiesWrite an inequality and solve:6. Anna needs to save at least $200 in the next 8 months so that she can have enoughto spend on her trip to Europe. She has $48 saved already. How much does she needto save each month to make sure that she will have enough spending money?
inequality:________________
solution: _______
7. Carly is driving to Baltimore. She leaves at noon and needs to arrive before 8pm. Ifthe entire trip is 416 miles, how fast must she average on the way there?
inequality:________________
solution: _______
8. Nicole is getting married and needs to figure out how many people she can afford toinvite. The cost per guest is $40 including dinner, and the location she has picked outcostss $1,250 to rent. If she wants to spend no more than $4,800 on her wedding,how many guests can she afford to invite?
inequality:________________
solution: _______
9. Tyreese is the quarterback on his football team and needs to pass for at least 2,250yards to break the record for his league. So far he has thrown for 1,410 yards. If thereare 6 games left in the season, how many yards does he need to throw for in eachgame to guarantee he will break the record?
inequality:________________
solution: _______
10. Parker is 19 months old and growing really fast. He now weighs 28 pounds. Hisparents don’t want him to be obese, so he should weigh less than 31 pounds at age 2.How many pounds can he gain per month?
inequality:________________
solution: _______
AlgebraInequality Word ProblemsName________________________ Period _____
2.5+Simple InequalitiesWrite an inequality and solve:1. Between 14 and 20 percent of Americans surveyed have blue eyes. This number isonly three percent more than half what it was 100 years ago. What percent of Ameri-cans had blue eyes 100 years ago?
compound inequality:________________
solution: _______
2. The human body contains about 18 times as many bacteria cells as human cells. Ifestimates range from 540 to 900 trillion bacterial cells in the body, how many humancells are in a typical person (in trillions)?
compound inequality:________________
solution: _______
3. Zach is learning to ride a bicycle. When his parents decide to help him buy a bike,they offer to pay $100 towards the purchase, plus they will match whatever amount hespends of his own money. The bikes Ethan like all cost between $230 and $450. Howmuch will Ethan need to spend of his own money to buy a bike he wants?
compound inequality:________________
solution: _______
4. Every week, Taha does between 875 and 1,050 pushups. If he does the samenumber of pushups each day every week, how many does he do each day?
compound inequality:________________
solution: _______
5. To convert from Celsius degrees to Fahrenheit, you multiply the Celsius temperatureby nine-fifths and then add 32. If the high temperature for tomorrow is going to begreater than 68 but less than 77 degrees Fahrenheit, what will the temperature be inCelsius?
compound inequality:________________
solution: _______
AlgebraTest Review (5, 7)Name________________________ Period _____
2.5+Solve:1. The largest angle in a triangle is twice the measure of the smallest, and the mediumangle is twenty-five degrees smaller than the largest angle. What is the measure of themiddle angle?
equation:________________
solution: _______
2. The length of a rectangle is six inches less than half its height. If its perimeter is 36inches, what is its area?
equation:________________
solution: _______
3. Danica’s last five math scores have all been at least 84 but no more than 96.Ashley’s scores have always been ten points more than twice Paula’s scores. Write andsolve a compound inequality to represent Paula’s scores.
compound inequality:________________
solution: _______
4. One number is three more than half another number. The sum of the numbers is15. Find their product.
equation:________________
solution: _______
5. The gas tank in Mr. Batterson’s car is three gallons larger than the tank in his wife’shybrid SUV. The car gets 24 miles per gallon, but the SUV gets 30 miles per gallon. Ifthe two vehicless can go the same distance on a tank of gas, how many miles can eachgo on a full tank?
equation:________________
solution: _______
6. Jack spends all of his allowance every week on candy. Last week he bought 3 candybars and spent the remaining $2.75 on gum. This week he bought five candy bars andhad $1.25 left for a big bag of Skittles. How much allowance does Jack get everyweek?
equation:________________
solution: _______
AlgebraTest Review (5, 7)Name________________________ Period _____
2.5+Mixed Review: Solve the following. Simplify Fraction answers.
Absolute Value Equations:
7. 7172 x 8. 7332 x
Absolute Value Inequalities: Graph each answer on a number line.
9. 93
7
x10. 12152 x
Radical Equations: Graph each answer on a number line.
11. 16732 x 12. 83
14
x
Proportions:
13. 4
425
7
xx14.
234
x
Formulas:
15. d
bcxa 2
16. cbxa )(
Compound Inequalities: Graph each answer on a number line.
17. xx
5127 18.
54
65
21
32
x
AlgebraTest Review (4)Name________________________ Period _____
2.5+Solve:1. The largest angle in a triangle is twice the measure of the smallest, and the mediumangle is twenty-five degrees smaller than the largest angle. What is the measure of themiddle angle?
equation:________________
solution: _______
2. The length of a rectangle is six inches less than half its height. If its perimeter is 36inches, what is its area?
equation:________________
solution: _______
3. Paul could lift a lot more before he hurt his leg. Now he can only lift 20 poundsmore than half of what he used to lift. If he can lift 150 pounds now, how much couldhe lift before he injured himself?
inequality:________________
solution: _______
4. One number is three more than half another number. The sum of the numbers is15. Find their product.
equation:________________
solution: _______
5. Jasmine needs to collect $40 to buy the costume accessories she wants for Hallow-een. She has collected $12 so far. If she sells candy corn for $0.70 a bag, how manybags does she need to sell if she wants to raise enough money for Haloween?
equation:________________
solution: _______
6. Julian spends all of his allowance every week on candy. Last week he bought 3candy bars and spent the remaining $2.75 on gum. This week he bought five candybars and had $1.25 left for a big bag of Skittles. How much allowance does Julian03. get every week?
equation:________________
solution: _______
AlgebraTest Review (4)Name________________________ Period _____
2.5+Mixed Review: Solve the following. Simplify Fractional answers.
Multi-Step:
7. 145
2
x8. 12)5(2 x
Absolute Value Equations:
9. 7172 x 10. 7332 x
Radical Equations:
11. 16732 x 12. 83
14
x
Proportions:
13. 4
425
7
xx14.
234
x
Compound Inequalities: Graph each answer on a number line.
15. xx
5127 16.
54
65
21
32
x
Both Sides:
17. 11553 xx 18. 153 xx
AlgebraPractice Test (5, 7)Name________________________ Period _____
3.6Solve for x: Simplify fractional answers.
1. 35
32
x
1. x=_______________
2. 735 xx2. x=_______________
3. xx3
565
3. x=_______________
4. 4112 x4. x=_______________
5. 4822 x5. x=_______or________
6. ad
cxy
6. x=_______
Solve for x: graph your answer on the number line..
7. xx 5)3(4 7. ____________________________
8. 12852 x8. ____________________________
9. xx 52
912
9. ____________________________
AlgebraPractice Test (5, 7)Name________________________ Period _____
3.6Write an equation/inequality and solve:10. The sum of two integers is 45 and the difference between the same two numbersis 11. What is the greater of these two integers?
equation:_____________________
solution: _______
11. In the morning Alexis rides the bus to school averaging 30mph but on the wayhome (along the same route) in traffic and with stops the bus averages just 20mph. Ifit takes a half-hour longer to ride home than to ride to school, how far does she ride onthe bus each way to school?
equation:________________
solution:_______
12. The perimeter of a rectangle is 62cm. If the height is two inches less than twicethe width, what is the area of the rectangle?
equation:________________
solution:_______
13. What is the measure of the largest angle in the triangle below?
equation:________________
solution:_______
14. Mr. Lyons runs between 21 and 35 miles every week (7 days) along the samepaths. If he changes his route so that every day he runs an extra mile, how many mileswill he run every day. Both the equation and the solution should be in the form of acompound inequality. (ex. -7<2x+3<25 and -5<x<11)
equation:_____________________
solution:____________
2xo xo
4x+5o
AlgebraPractice Test (4)Name________________________ Period _____
3.6Solve for x: Simplify fractional answers.
1. 143 x1. x=_______________
2. 2125 xx2. x=_______________
3. 35
32 x
3. x=_______________
4. 735 xx4. x=_______________
5. xx3
565
5. x=_______________
6. 4112 x6. x=_______________
7. 4822 x7. x=_______or________
8. xx
532
8. x=_______
Solve for x: graph your answer on the number line.
9. xx 5)3(4 9. ____________________________
10. 72
3
x
10. ____________________________
AlgebraPractice Test (4)Name________________________ Period _____
3.6Write an equation/inequality and solve:10. The sum of two integers is 45 and the difference between the same two numbersis 11. What is the greater of these two integers?
equation:_____________________
solution: _______
11. The sum of consecutive odd integers is -28. What is the greater of these twointegers?
equation:________________
solution:_______
12. The perimeter of a rectangle is 62cm. If the height is two inches less than twicethe width, what is the area of the rectangle?
equation:________________
solution:_______
13. What is the measure of the largest angle in the triangle below?
equation:________________
solution:_______
14. Cameron is saving to buy his favorite video game system. He has saved $120 sofar. If he saves $15 a week, how many weeks will it take before he has enough to buythe $300 game system?
equation:_____________________
solution:____________
2xo xo
4x+5o
AlgebraProportions and Percents 4.3Percent means per hundred.Think of some words that contain the word cent:
Fractions can be used to represent ratios.
For example:
53
can be used to represent “three out of five”..
One of the easiest ways to change a ratio to a percent is by using aproportion.
Example: Express three out of five as a percent:
You may have learned the following:
100%
ofis
Is over of equals percent over 100.
This is the useful for very simple problems involving percents:
Examples: Use the Percent Proportion to Solve:
1. What percent is 12 of 40?2. 6 is 30% of what number?3. What is 20% of 45?
I prefer you use the following variation of the percent proportion:
100%
WholePart
The part over the whole equals percent over 100.
Examples: Use the Percent Proportion to Solve:
1. There are 60 words altogether on the vocabulary list. Jennifer knows65% of them. How many of the words does she know?
2. Lewis took 15% of the candy in the bag. If Lewis took 12 pieces, howmuch candy was there in the bag?
3. 19 of the fish are spotted. If there are 50 fish in the tank, whatpercent of the fish have NO spots?
AlgebraPercents and ProportionsSolve each using a proportion. Round decimal answers to the tenth.
____1. What percent is 7 of 40?
____2. What is 15% of 20?
____3. 12 is 60% of what number?
____4. Ryan made 7 out of 20 free-throw attempts. What percent is this.
____5. 90% of the chocolates sold were milk chocolate. If 80 chocolates were sold, how many weremilk chocolate?
____6. What percent is 15 of 75?
____7. There are 12 girls in the class. If 20% of the students in class are girls, how many boys arein the class?
____8. Six of the numbers on the list are odd numbers. If there are 16 numbers on the list, whatpercent of them are odd numbers?
____9. The football team has won 35% of its games in the past 10 years. If they have lost 52games, how many games have they won?
____10. 52% of the undergraduate students at UNC are female. If there are 6,292 female under-graduate students at UNC, how many undergraduates attend UNC all together?
____11. What percent of 90 is 15? Express your percent answer as a mixed number.
____12. Jon read 85% of the book assigned for homework. The book is 320 pages long. Howmany pages does Jon have left to read?
____Challenge 1. Carrie has won 7 of her first 15 tennis matches. How many wins does she needin a row to improve her winning percent to 75%?
____Challenge 2. John has a bag full of marbles. Fifty percent of the marbles in the bag are red,but if he adds three red marbles to the bag, 60% of the marvbles in the bag will be red. Howmany marbles are in John’s bag?
Name________________________ Period _____
AlgebraPercent Change 4.3Percent Change:
100
% changeoriginalchange
Change over original value equals %/100
Examples: Use the Percent Proportion to Solve:
1. The price on a shirt went from $15 to $12. What was the percent off?2. Cary has a population of about 100,000 residents, up from only 60,000just 15 years ago. By what percent has Cary’s population increased?3. Grant grew by 15% in the past two years. If he was 60 inches tall twoyears ago, how tall is he now?4. You must pay 9% sales tax on all prepared foods. If a Happy Mealcosts $3.25, what is the price after sales tax (rounded to the cent)?
More Percent Change:
100
%neworiginal
new New amount over the original equals new%/100.
This makes some problems much easier when you don’t know thechange. Try the previous set using this formula.
Examples: Use the Percent Proportion to Solve:
1. Best Buy decreased the cost of its best flat screen monitor by 20%,and it is now being sold for $429.99. What was the cost before the dis-count (to the cent)?2. Belle improved her fast pitch speed by 25%. If she was pitching48mph before, what speed can she pitch now?3. The population of Bobaloobaville increased by 8% last year. If thereare 21,060 people in Bobaloobaville now, how many were there a yearago?4. Mr. Lyons improved his mile time by 5%. Last year he could run themile in 6 minutes. What is his mile time this year?
AlgebraPercent Change 4.4More Percent Change:Why you need new over original:
The most common mistake that people make when solving percentchange problems occurs on questions like this:
Ex: Brandon weighs 10% more than Phillip. If Brandon weighs 220pounds, how much does Phillip weigh?
Many students get 198. Why!?!
Try solving these problems mentally:1. You have $100. You lose 10% of your money, then gain 10%. Howmuch do you have now?2. You have $100. You lose 50% of your money. What percent wouldyou need to gain to have $100 again?
This is why we need: 100
%neworiginal
new
Examples: Use the Percent Proportion to Solve:1. Mr. Batterson invested money in a stock that has increased in value by44% and it is now worth $5,760. What was the value of the original in-vestment?2. Deborah improved her quarter mile time by 6.6 seconds, and she nowruns the quarter-mile in 59.4 seconds. What percent improvement isthis?
Practice: Use the Percent Proportion to Solve:1. Find the original price on a refrigerator if you paid $588.49 after a 7%sales tax.2. Sarah improved 100m dash time from 12.21 seconds to 11.48 sec-onds. By what percent did her time decrease? (to the nearest percent)3. Antonio is trying to gain weight for football in high school. He weighs140 pounds now and wants to weigh 161 pounds before tryouts. Whatpercent of weight gain is this?4. Anna scored a 767 on her most recent math SATs. This is 18% betterthan her previous score. What was her original SAT score?
Challenge:Phillip weighs 20% less than Brandon. What percent would Phillip needto gain to weigh the same as Brandon?
AlgebraPercent Change Shortcuts 4.4Decimals and PercentsTo convert a percent to a decimal, move the decimal point two places tothe left.Easy:25% = _____ 14% = _____ 6% = _____Harder:25,000% = _____ .014% = _____ 6.06% = _____
Try using is over of on the following three problems:
1. What percent of 340 is 51?2. 51 is 15% of what number?3. What is 15% of 340?
For #3 there is an easier way:To find a percent of a number, convert the percent to a decimaland multiply.
Ex. What is 12% of 180?
Practice:1. What is 25% of 190?2. What is 10% of 34?3. What is 3.5% of 650?4. What is 450% of 19?
This is especially helpful in problems involving a percent increase or de-crease:
Ex. Find the price after tax on the following items using the given tax.Round to the cent.
1. $40.00 (5%) 2. $22.50 (7.5%) 3. $314.99 (3.9%)
Practice:1. A volleyball is being sold for 25% off. If the original price was $15.96,what is the sale price?2. The cost of gas increased last week by 2%. If the original cost was$2.50/gallon, what is the new cost?Challenge: A textbook you need for college has been reduced in priceby 30%, and you have a coupon which allows you to save an additional25%. If tax is 5%, how much will you pay for the book which was origi-nally $40?
AlgebraPercents and ProportionsName________________________ Period _____
4.4Solve. Use the skills we have learned so far to answer the following:
1. What number is 22% of 280?1. _______
2. What percent is 123 of 164?2. _______
3. What number is 12% more than 425?3. _______
4. 20% less than a number is 76. What is the number?4. _______
5. Tax on a $60.95 pair of shoes is 5%.How much will you pay after tax?
5. _______
6. Corey has 22% more money than Carla. If Corey has $30.50, howmuch money does Carla have?
6. _______
7. Sears marked up all washer/dryer prices by 8%. If the original priceon a Kenmore washer was $350, what is the price after markup?
7. _______
8. Apples are on sale: buy four get one free. This is the same as gettingwhat percent discount?
8. _______
9. After spending $8.40 on lunch Kayla now has 88% of the cash left inher purse. How much money did she start with?
9. _______
10. Jeremy has 20% more money than Sue, who has 20% less moneythan Richard. If Jeremy has $57.60, how much does Richardhave? (hint: correct answer is a whole dollar amount)
10. _______
AlgebraPercents and ProportionsName________________________ Period _____
4.4Solve. Find the cost of each item after the discount and/or tax.
Round answers to the cent.
11. Skis: $248.90 12. Envelopes: $2.95Tax: 7% Tax: 5%
Price: _________ Price: _________
13. Sofa: $598.95 14. Cereal: $4.89Tax: 10% Tax: 5.5%
Price: _________ Price: _________
15. Computer: $875.55 16. Dining Table: $185Discount: 10% Discount: 40%Tax: 3% Tax: 8%
Price: _________ Price: _________
17. Television: $428.99 18. Breakfast: $14.18Discount: 20% Tax: 4.5%Tax: 6% Tip: 20%
Price: _________ Price: _________
AlgebraPractice: Dilations on the PlaneGraphing figures on the coordinate plane is simple.Graph and connect each set of points below separately.Connect each set in order and then connect the first and lastpoints in each set.
1. (2,3) (9,3) (9,7) (2,7)What shape is this?What quadrant is it in?Find its area.
2. (-3,-3) (-7,-2) (-8,6) (-4,7)What shape is this?What quadrant is it in?Find its area.
3. (-7,-7) (-3,-9) (6,-6) (2,-4)What shape is this?What quadrant is it in?Find its area.
A dilation is a reduction or enlargement of the original figure. To createa dilation on the coordinate plane, multiply each coordinate by a scalefactor.Practice:Graph the following triangle.(-1, 2) (2,1) 3, -2)Find its area.
Dilate the original trianglewith a scale factor of 2.Find its area.
Dilate the original trianglewith a scale factor of 2.5.Find its area.
Predict the area of a dilationof the original with a scalefactor of 5.
4.4+
AlgebraPractice: Dilations on the PlanePlot each set of points and the dilations listed and answer thequestions that follow.
1. (0, -2) (2,2) (4,-2)
Find its area. _______
2. Graph a dilation with ascale factor of 2.
Find its area. _______
3. Graph a dilation with ascale factor of 2.5
Find its area. _______
4. Predict the area of adilation with a scale factor of 10.
________
Name________________________ Period _____
5. (-3, 1) (1,1) (2,-2) (-2,-2)
Find its area. _______
6. Graph a dilation with ascale factor of 2.
Find its area. _______
7. Graph a dilation with ascale factor of 3.
Find its area. _______
8. Predict the area of adilation with a scale factorof 6.
________
4.4+
AlgebraDilations on the PlaneDetermine the scale factor used to dilate each pair.Some answers may be fractions.
9. a to b _____
10. b to a _____
11. f to c _____
12. f to d _____
13. e to c _____
14. c to d _____
a
b
c
de
f
Complete the following dilations below.
15. Original:(-4,4) (-8, -8) (8,-4)
16. Dilation with a scale factorof 1/4.
______________________
17. Dilation with a scale factorof 0.75.
______________________
18. Dilation with a scale factorof 1.2
______________________
Name________________________ Period _____
4.4+
Algebra 4.4+Similar TrianglesProportions and Similar Triangles:Similar triangles are the SAME SHAPE not the same size.Corresponding angles are equal.Corresponding sides are proportional.You can use proportions to find the length of missing sides.
Ex. Find the length of sides x and y of the similar triangles below:
2.4
1.8
3.6 5.4
y
x
1628
23x
y
20
1. 2.
Practice. Find the length of sides x and y of the similar triangles below:Round to the tenth.
1. 2.
3. 4.
y
x12.5
21.59.1
17.0
y
x
x
x
y
y
25 30
16.5
22.5
6.3
2.7
3.6
7.6
10.5
13.5
8.4
12.6
AlgebraProportions ReviewSolve each:
100 What is 20% of 15?
200 You need to get 70 percent of the questions right on a test to pass.How many questions can you miss on a 60-question exam and still pass?
300 What is 30% more than 70?
400 The average gas mileage for small automobiles in America has im-proved by 15% in the past decade and is now 27.6 mpg. What was theaverage gas mileage 10 years ago?
500 Marcia got a 10% raise in 2005, and a 15% raise in 2007. She nowmakes $44,275 a year, what was her salary before her two raises?
Find each missing length in the similar figures below:
35
62
x (100)
y(200)
38
5
(500) x
37
2
6
(300) x
y (400)
Algebra
b
Proportions Review
c
a
Find each scalefactor:
(100) a to b
(200) a to c
(300) b to c
(400) c to a
(500) c to b
On your graph board with a straight-edge:
(100) Plot and connect (0,2) (6,0) and (2,-4)
(200) Plot a dilation with a scale factor of 1/2
(300) Plot a dilation of the original with a scale factor of 2
(400) Plot a dilation of the original with a scale factor of 3/2.
(500) Plot a dilation of the original with a scale factor of 5/3.
AlgebraPractice Quiz: ProportionsName________________________ Period _____
4.4+Find the length of the missing sides x and y. Round to the tenth.
1.
1. x_______ y _______
2.
2.x_______ y _______
Solve: Round decimal answers to the tenth.Use bar notation for repeating decimals. Show all units.
3. What is 15% of 150?3._____
4. 10 is what percent of 290?4._____
5. 20 is 125% of what number?5._____
6. What number is 40% of 12?6._____
7. What number is 15% greater than 80?7._____
8. 60% less than a number is 33.6. What is the number?8._____
9. What number decreased by 70% equals 15?9._____
x
8
15.4y
11.2
y
9
14 9.1 11.7
x
9
AlgebraSolve: Round decimal answers to the tenth unless noted otherwise. Show all units.
10. A coat is on sale for 30% off. If the original price was $56.95, what is the sale price tothe nearest cent?
10._____
11. The tax on a $10.95 pizza is 9%. How much change will you get back if you pay with a$20 bill?
11._____
12. The number of students at Ligon decreased by 6% this year. There are 1,021 studentsat Ligon this year, how many were here last year (to the nearest student).
12._____
13. A $65.00 stereo costs $69.55 after tax is added. What percent is the tax amount?13._____
14. A calculator is on sale for 25% off and now costs $61.74. What was the original priceof the calculator to the nearest cent?
14._____
Find the percent of increase or decrease. Round to the tenth of a percent.
15. New price: $24.90Original Price: $22.91
15._____
16. New price: $125.00Original Price: $320.00
16._____
Determine the scale factor used in each dilation below:
17-20.17. a to b_____
18. a to c_____
19. d to e_____
20. e to d_____
ab
c
d
e
Practice Quiz: ProportionsName________________________ Period _____
4.4+
AlgebraInterpreting Function GraphsUnderstanding and interpreting graphs can be difficult.Uderstanding the relationship between the x and y-axis is veryimportant.
Example: The following graph shows the distance traveled by a schoolbus based on every morning from 6:30-7am.
dista
nce (
mile
s)
time (minutes)0 15 30
10
20 1. What do the flat sections of the graphrepresent?
2. What do the steep sections of the graphrepresent?
3. Which is the best estimate of the busesaverage speed: 17mph, 20mph, 34mph,50mph?
Why doesn’t path of the line ever go down?
Often you will need to recognize graphs plotting distance versus time, ordistance from a point versus time, usually with time plotted on the x-axis.
Distance is called a Function of TimeMatch each graph below with the appropriate situation:1. 2. 3.
0 0 0time time time
dista
nce f
rom
hom
e
dista
nce f
rom
hom
e
dista
nce f
rom
hom
e
a. Mary leaves her house, walks to the store, and returns with groceries.
b. Roger walks around the block.
c. Sylvia runs to the track, takes a few laps, and jogs home.
AlgebraInterpreting GraphsUse the two graphs below to answer the questions that follow:
tota
l dist
ance
(mile
s)
time (minutes)0 20
10
20
30
Dan’s Drive to Work
1. When is Dan driving on the highway?2. Approximate Dan’s average speed for the trip to work.3. Approximate Dan’s top speed during the trip.4. Who finishes the marathon first?5. How do Megan and Carl’s strategies compare?6. Approximately how many minutes does it take Carl to run each
mile during the first 1:40 of the race? (4, 5, 6, or 7 minutes?)
30
10
tota
l dist
ance
(mile
s)
time (hours)0 2
10
20
Marathon RunnersMegan Carl
25
1 3
15
5
These graphs look very different from graphs representingspeed versus time.
tota
l dist
ance
(mile
s)
time (minutes)0 20
10
20
30
Dan’s Drive to Work30
10
spee
d (m
ph)
time (minutes)0 20
25
50
30
Dan’s Drive to Work75
10
Use a sheet of graph paper to graph the following:1. The distance you are from your locker from 7:20-12:20.2. The total distance you have traveled during the same time period.3. The speed you are moving during the same time period.
Be prepared to be able to explain your three graphs.
AlgebraFunctions 5.2Functions: Write this down and memorize it!
y is a function of x only if:for every domain value (x)there is exactly one range value (y).
This is easiest to see on a graph using the Vertical Line Test:
Examples: Which of the following are functions?
Examples: It is also easy to tell if a Relation (set of points) represents a function. Which is a function? Why or why not?
1. )0,1)(1,1)(2,3)(4,5( 2. )5,6)(4,5)(3,6)(2,5(
Practice: Determine which relation represents a function:
1. )7,6)(5,4)(4,3)(3,2( 2. )3,4)(3,3)(3,2)(3,1(
3. )2,2)(2,2)(2,2)(2,2( 4. )90,8)(70,6)(50,4)(30,2(
Examples: It is more difficult to determine if an equation is a function for y:Explain why each equation belongs in the category it is in.hint: Try plugging-in 4 for x in each equation,
see how many answers are possible for y.
Functions: Not Functions:
32 yx 32 xy 2yx 4x
3 xy 20xy yx xy
AlgebraFunctions 5.2Review:y is a function of x if:Each x gives one value for y.The graph passes the vertical line test.
Complete:Every linear equation is a function unless its slope is ________.List an example of a linear equation that is NOT a function.
Function Notation:
If y is a function of x, y can be replaced with f(x).
Examples:
1. 32 xy means exactly the same thing as ______________.
2. xxy 32 means exactly the same thing as ______________.
Do not get confused by function notation, if you get lost...switch f(x)= back to y=.
In function notation, questions are often asked like this:
321)( xxf Find )4(f : This is a fancy way of saying plug 4 in.
To denote different functions, you may often need to use )(xg or )(xh .
Practice:Find the range of each function for the given domain:
1. 75)( xxf {D: -3, -2, -1, 0}
2. 2)( 2 xxg {D: 9, 7, 5, 3}
Challenge. Using both functions above, find )2(gf and )2(fg .
AlgebraFor each pair below, answer whether each is a function.Write A. B. Both or Neither in the blank provided.
1. A. 103 yx B. 7y
1. ____________________
2. A. (2,2) (3,3) (4,4) (5,5) B. (2,1) (3,1) (4,1) (5,1)
2. ____________________
3. A. xy B. yx 3. ____________________
4. A. yyx 2 B. xxy 2
4. ____________________
5. A. (0,1) (1,2) (2,3) (3,4) B. (0,-1) (1,-2) (2,-3) (3,-4)
5. ____________________
Use the functions below to answer the questions that follow:
5)( xxf 22)( xxg xxh
21)(
6. )3(f _______ 7. )4(f _______
8. )2(g _______ 9. )3(g _______
10. )5(h _______ 11. )0(h _______
12. )2()2( gf _______ 13. )3( xg _______
14. )5(xh _______ 15. )2(hg ______
Find the range for the domain given:
15. 32)( xxf }7,5,5,7:{ D15. ____________________
Challenge. 52)( 2 xxf }55:{ xDC. ____________________
Practice: FunctionsName________________________ Period _____
AlgebraFunctions 5.4Tables and Functions:You can create a table of values for a function just as youwould any other equation.
Review:
Create a table of values for the function 75)( xxf
x f(x)
You can go in reverse as well. Some functions are obvious.Practice:Try to determine what function was used to create each table:1. 2. 3.
f(x)= f(x)= f(x)=
Other times, it is not as easy.For linear equations there is a method that will always work.1. Find the slope. 2. Use Slope-Intercept Form and solve for b.
Examples:Try to determine what function was used to create each table:1. 2. 3.
f(x)= f(x)= f(x)=
x f(x)2 123 134 14
x f(x)5 107 149 18
x f(x)-2 2-7 7 9 -9
x f(x)2 -95 -156 -17
x f(x)-6 -6-8 -7 4 -1
x f(x)4 16-1 1-3 9
AlgebraFunctions 5.4Practice: Write a function for each table of values.
1. 2. 3.
f(x)= f(x)= f(x)=
x f(x)-3 -21 -65 -10
x f(x)6 -13 -3-9 -11
x f(x)7 492 4-3 9
Some functions will not be linear.If the slope of an equation is not constant, the function is not linear.
Look for squares, square roots, absolute value, and other common functions.
Practice: Write a function for each table of values.
1. 2. 3.
f(x)= f(x)= f(x)=
x f(x)1 -14 -29 -3
x f(x) 3 3-4 4-6 6
x f(x)1 12 0.54 0.25
Practice: Be careful, tables will not always look the same.
1. 2. Three miles 3.in two hours.
Five miles inthree hours.
Seven miles in f(a)=four hours.
f(x)= f(h)=
f(x) x-5 3-9 53 -1
a 4 -4 -8f(a) -4 2 5
AlgebraFunctionsWrite a function for each table of values.
1. 2. 3.
f(x)= f(x)= f(x)=
x f(x)5 97 139 17
x f(x)-3 10-1 81 6
x f(x)-7 14-3 6 9 -18
4. 5. 6.
f(x)= f(x)= f(x)=
f(x) x3 3-5 7-13 11
x f(x)-9 0-6 23 8
x f(x)-3 -48 7 -11 -12
7. 8. 9.
f(x)= f(x)= f(x)=
x f(x)5 -8-5 -10-15 -12
x f(x)4 -1.83 -2.12 -2.4
x f(x)5 12.511 27.5 -1 -2.5
Name________________________ Period _____
5.4
Write a function for each table of values. These are not linear.
10. 11. 12.
f(x)= f(x)= f(x)=
x f(x)-2 55 2612 145
x f(x)-3 2-7 69 10
x f(x)16 -49 -3 25 -5
AlgebraPractice:Try to determine what function was used to create each table:1. 2. 3.
f(x)= f(x)= f(x)=
Functions can also be written to describe many real-world problems:
Example:Write a function that describes the cost to rent a widescreen television for theSuperbowl if it costs $20 plus $14.50 a day. Cost is a function of ________.
Harder Example:Write a function that describes the cost of gas if your car gets 30mpg and gascosts $3 a gallon. Cost is a function of ___________.
Practice:1. Write a function that describes how far you can drive going 50mph.Distance is a function of _________.
2. Write a function that describes the profit you make selling basketball ticketsfor $4 each. Profit is a function of ___________.
3. Write a function that describes the profit you make selling basketball ticketsfor $6 each if you already spent $100 advertising the basketball game. Profit isa function of ___________.
Practice:1. Write a function that describes your math test score if you lose 6 points forevery wrong answer (starting at 100). Score is a function of ____________.
2. Write a function that describes the change you will get back buying multiple$0.50 candy bars with a $10 bill. Change ($) is a function of ___________.
3. Write a function that describes the cost to have your vehicle towed if youare charged $15 for the pickup plus $2 for every mile.
Functions 5.4
x f(x)5 -93 -5-1 3
x f(x)9 -53 -9-9 -17
x f(x) 8 4 0 0-4 -2
AlgebraQuiz Review 5.4Which of the following ARE functions(100 points each, -100 for every one you get wrong):
A. (1,1) (2,2) (3,3) (4,4) B. (1,2) (2,1) (3,4) (4,3)
C. (-1,4) (-2,4) (-3,4) (-4,4) D. (-4,1) (-4,2) (-4,3) (-4,4)
E. (-2,4) (2,-4) (4,-2) (-2,4) F. (5,10) (-5,10) (10,5) (-10,5)
Which of the following ARE NOT functions(100 points each, -100 for every one you get wrong):
A. B.
C. D.
Which of the following ARE functions(200 points each, -200 for every one you get wrong):
A. 12 xy B. xy 1
C. xy 2)1( D. 2xy
AlgebraQuiz Review 5.4Use the following functions to solve each:
32)( xxf 1)( 2 xxg
100. )3(f 200. )3(g
300. )3()3( gf 400. )2(fg
Write a function for each:
100. 200.
300. 400.
Write a function for the following:
200. The score you get on a test if you lose 7 points for every wrong answer.
200. The cost of a canister of cashews if the charge is $5.50 a pound plus$0.50 for the canister.
200. The cost of a gym membership if there is a $50 enrollment fee andthe charge is $35 a month.
x f(x)2 -15 28 5
x f(x)-3 -2-6 -6 9 14
x f(x)3 -17 1-5 -5
x f(x)-8 9-2 3 1 2
AlgebraFor each pair below, answer whether each is a function.Write A. B. Both or Neither in the blank provided.
1. A. 1052 yx B. 1 xy
1. ____________________
2. A. (1,2) (0,3) (-1,2) (-2,3) B. (2,1) (3,0) (2,-1) (3,-2)
2. ____________________
3. A. B.
3. ____________________
4. A. 2yx B. 2xy 4. ____________________
5. A. (0,1) (1,2) (2,3) (3,4) B. (0,-1) (1,-2) (2,-3) (3,-4)
5. ____________________
Use the functions below to answer the questions that follow:
32)( xxf )3()( xxxg xxh 3)(
6. )3(f _______
7. )2(g _______
8. )0(h _______
9. )2()2( gf _______
10. )5()5( gg _______
Practice Quiz: Functions 5.4Name________________________ Period _____
AlgebraWrite a function for each table of values below:
11.
11. f(x)=____________________
12.
12. f(x)=____________________
13.
13. f(x)=____________________
14.
14. f(x)=____________________
15.
15. f(x)=____________________
Write a function for each and answer the questions that follow:
16. The distance traveled by a train traveling at 80mph.Distance is a function of time in hours.
16. ____________________
17-18. The total amount spent at the fair if rides cost $1.50 and theentrance fee is $7.50.
17. Cost is a function of ____________________
18. Write the function ____________________
19-20. The profits from a raffle if tickets are sold for $5 and $245.95 wasspent to set up the raffle.
19. Write the fiunction ____________________
20. How many tickets must be sold to guarantee you make at least $100 profit? ____________________
Practice Quiz: Functions 5.4Name________________________ Period _____
x f(x)5 13 -3-1 -11 x f(x)
4 56 68 7 x f(x)
-3 -3-6 -1-9 1x f(x)
5 011 -12-2 14 x f(x)
16 4 9 3 25 5
AlgebraIntro to Linear Equations 6.0Linear Equations:
72 xy 521
xy 1232 yxLinear Equations generally contain two variables: x and y.In a linear equation,y is called the dependent variable andx is the independent variable.This is because y is dependent on what you plug-in for x.The domain of a linear equation is the set of all x-coordinates and therange is the set of all y-coordinates.
Examples:State the range and the domain for each set of points below.
1. (-3, -3) (-1, 1) (1, 5) (3, 9) (5, 13)
2. (-5, 8) (-2, 5) (1, 5) (4, 9) (7, 13)
Practice:State the range and domain for each set of points graphed below as aninequality:Ex. #1 Domain -4 < x < 4 Range: 5 < y < 9
1
23
4
5
AlgebraIntro to Linear Equations 6.0Given a domain, it is easy to find the range for any linear equation.
Examples:Find the range for the given domain:
1. 32 xy {D: -3, -2, -1, 0}
2. 32 xy {D: 6, 1, -4, -9}
Practice:Find the range for the given domain:
1. 73 xy {D: -3, -2, -1, 0}
2. 132
xy {D: 6, 3, 0, -3, -6}
3. 1062 xy {D: -1, 1, 3, 5}
In problems like #3 above, it helps a lot to solve for y before plugging invalues for the domain.
Practice:Find the range for the given domain. Begin by solving for y.
1. 5472 xy {D: -3, -2, -1, 0}
2. xy 963 {D: 6, 3, 0, -3}
3. 1035 xy {D: x>5 }
AlgebraIntro to Linear EquationsName________________________ Period _____
6.0For each graph below, state the domain and range using an inequality:
1. Domain: _____________
Range: _____________
2. Domain: _____________
Range: _____________
3. Domain: _____________
Range: _____________
4. Domain: _____________
Range: _____________
1
2 3
4
List the Domain and Range for each set of points listed below:
5. (9,1) (8,2) (7, 3) (6, 4) (5, 5)5. Domain: _____________
Range: _____________
6. (-3, -3) (-3, -4) (-3, -5) (-3, -6)6. Domain: _____________
Range: _____________
7. 3 xy for x = -3, -4, 5, and 6
7. Domain: _____________
Range: _____________
8. 52 xy for x > 3
8. Domain: {D: x > 3}____
Range: _____________
AlgebraIntro to Linear EquationsName________________________ Period _____
6.0Given each domain below, find the range for each equation. Solve for y where necessary.
9. 32 xy {D: -1, 0, 1, 2}
9. Range: _____________
10. 521
xy {D: -4, -2, 0, 2}
10. Range: _____________
11. 1263 xy {D: -5, -3, 1, 5}
11. Range: _____________
12. 4 yx {D: 4, 1, -1, -9}
12. Range: _____________
13. 63 xy {D: -3, 0, 3, 9}
13. Range: _____________
14. 1035 xy {D: -15, -10, -5, 0}
14. Range: _____________
15. 732 xyx {D: -4, 4, 12, 20}
15. Range: _____________
AlgebraGraphing A Linear Equation 6.0To graph a Linear Equation:1. Solve for y.2. Setup a table of x and y values.3. Plot at least three coordinates and connect them.
Ex.Graph
72 xy
Graph
232
xy
PracticePlot each of the following equations on the same graph.
1. 43 xy
2. 5
43
xy
3. 93 xy
AlgebraGraphing A Linear Equation 6.0PracticePlot each of the following equations on the same graph.
1. 5 yx
2. 923 xy
3. xy 2123
PracticePlot each of the following equations on the same graph.
1. 53 yx
2. yx 43
3. )3(325 xy
AlgebraGraphing Linear EquationsName________________________ Period _____
Graph each equation below on the graphs provided.
1. 9 xy 2. 1223 xy
3. 426 yx 4. )6(213 xy
6.0
AlgebraGraphing Linear EquationsName________________________ Period _____
Graph each equation below on the graphs provided.
5. 13114 xy 6. xyx 362
7. yx 2
43
8. 5y
6.0
AlgebraStandard Form 6.0Standard Form of a linear equation:
62 yx 2173 yx 162 yx
Examples above are Linear Equations written in Standard Form.Here is Standard Form. MEMORIZE THIS.
CByAx 1. No absolute value, exponents, square roots, etc.2. 1 or 2 variables (A and B cannot both be zero).3. All linear equations can be written in Standard Form.4. A, B, and C are Integers (not fractions). A should be positive.
Practice:Label the values for A, B, and C in each linear equation below.
1. 62 yx 2. 2173 yx 3. 7x
Examples:Convert each equation below into Standard Form if possible.Get both variables ON THE SAME SIDE OF THE EQUATION.
1. 53 xy 2. 43
32
21
xy 3. yx 53
Practice:Convert each equation below into Standard Form if possible.
1. xy 5 2. xy41
53
3. yxx )1(
AlgebraStandard FormName________________________ Period _____
6.0Convert each equation below into Standard Form.Remember to remove all fractional coefficients.
1. 3 yx 2. xy 273
3. yx
4
74. yx 1022
5. 7x 6. 122 y
7. yx 4 8. xy 25
9. 341
32
yx 10. 251
21
yx
11. 732
61
yx 12. yx81
43
AlgebraStandard FormName________________________ Period _____
6.0Each Equation below is written in Standard Form.Solve each for y, create a table of values, and graph each.
13. 3 yx 14. 1236 yx
15. 63 yx 16. 62 yx
AlgebraStandard Form and Intercepts 6.0On a graph, the x-intercept is where the line crosses the x-axis.The y-intercept is where a line crosses the y-axis.
Practice:Look at the graphs below and give the coordinates ofthe x and y-intercepts.
1.x-int. _______
y-int. _______
2.x-int. _______
y-int. _______
3.x-int. _______
y-int. _______
notes:The x-intercept always occurs where y equals _____.The y-intercepts always occurs where x equals _____.
Set y=0 to find the x-intercept.Set x=0 to find the y-intercept.
Examples: Find the x and y-intercepts of each.We will call this the coverup method.
1. 123 yx 2. 452 yx 3. 832 yx
Practice: Find the x and y-intercepts of each.
1. 3035 yx 2. 117 yx 3. 732
43
yx
1
2
3
AlgebraStandard Form and Intercepts 6.0Practice:Each line below goes with one of the linear equations on the left.Match each equation with its graph by finding the intercepts.
1. 1232 yx
2. 1053 yx
3. 73 yx
Practice: Graph each of the following using the intercepts:
1. 1535 yx
2. 82 yx
3. 93 yx
A
B
C
AlgebraStandard Form and Intercepts 6.0Determine the x and y-intercepts for each equation below.Convert to Standard Form where necessary.
1. 93 yx 2. 1025 yxx-int.: _______ x-int.: _______
y-int. ________ y-int. ________
3. 79 yx 4. 372 yxx-int.: _______ x-int.: _______
y-int. ________ y-int. ________
5. yx 39 6. 523 xyx-int.: _______ x-int.: _______
y-int. ________ y-int. ________
7. 32 xy 8. yx 4122
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
9. xy
3
210.
543
xy
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
11. 59
xy
12. 31
92
21
xy
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
Name________________________ Period _____
AlgebraStandard FormName________________________ Period _____
6.0Graph each equation below using the intercepts.Connect the intercepts. Intercepts are all whole numbers.
13. 62 yx 14. 1236 yx
15. 63 yx 16. 102 yx
AlgebraSlope 6.1The Slope of a line is its RISE over RUN.1. Read graphs left to right, just like sentences.2. Find a point on the graph of a line.3. Count how far you must go UP AND OVER to get to the next point.4. Write this as a fraction: Ex.
Examples: State the slope of each line:
32
32
overupSlope
1
2
3
AlgebraSlope 6.1Slope is not always positive. Working from left to right,if you go down and over, this is negative slope.
Examples: State the slope of each line:
Practice: State the slope of each line:
1
2
3
1
2
5
4
3
AlgebraSlope 6.1You do not need a graph to find the slope of a line.How could you find the RISE given two coordinates? ex. (4, 2) and (8, 10)How could you find the RUN given two coordinates?Example:Find the slope of the line passing through (3, 5) and (7, 7).How far UP?How far OVER?
notes:
Given two coordinates: ),( 11 yx and ),( 22 yx
Slope Formula: 12
12
xxyym
memorize this!
To find the slope you must divide the y’s and the x’s.y minus y over x minus x.Rise over run.That’s how you find the slope.
Examples:Find the slope of a line passing through each given pair of points:
1. (9, 4) (7, 10) 2. (-2, -5) (4, 1)
Practice:Find the slope of a line passing through each given pair of points.Simplify all slopes and LEAVE IMPROPER FRACTIONS:
1. (-3, 0) (1, 2) 2. (3, 4) (4, -1)
3. (6, -2) (7, -7) 4. (1, -5) (-9, 1)
5. (8, 4) (-5, 15) 6. (-1, -5) (4, -10)
AlgebraSlope-Intercept Form 6.2Graphing a Linear Equation:Method 1: x/y ChartMethod 2: Intercepts (from Standard Form)
Use one of the methods above to graph each of the following equations.Then, list the slope and the y-intercept of each equation.
521
xy
slope: _____
y-int: _____
332
xy
slope: _____
y-int: _____
Guess what form we are going to learn next....
Slope-Intercept FormMEMORIZE THIS:
bmxy Where m is the slope and b is the y-intercept.
This is the most useful form of a linear equation, especially for graphing.
AlgebraSlope-Intercept Form 6.2Practice: Graph each using Slope-Intercept Form.
1. 532
xy 2. 52 xy 3. 1253 xy
Practice: Convert each into Slope-Intercept Form, then graph.
1. 217 yx 2. 303 yx 3. 255 yx(Why is Standard Form less useful for graphing these equations?)
Practice: Write an equation for each line graphed belowin Slope-Intercept Form.
Practice: Convert these answers to Standard Form.
1. 254
xy 2. 631
xy 3. 332
xy
1
2
3
AlgebraSlope-Intercept Form 6.2Convert Each into Slope-Intercept Form
1. 93 yx 2. 1025 yx
slope: _______ slope: _______
y-int. ________ y-int. ________
3. 189 yx 4. 2172 yx
slope: _______ slope: _______
y-int. ________ y-int. ________
5. yx 39 6. 1523 xy
slope: _______ slope: _______
y-int. ________ y-int. ________
7. 3025 xy 8. yx 4122
slope: _______ slope: _______
y-int. ________ y-int. ________
9. xy
3
210.
643
xy
slope: _______ slope: _______
y-int. ________ y-int. ________
11. 59
xy
12. 31
92
21
xy
slope: _______ slope: _______
y-int. ________ y-int. ________
Name________________________ Period _____
AlgebraSlope-Intercept FormName________________________ Period _____
6.2Graph each equation below using slope-intercept form.
13. 5
32
xy14. 1835 yx
15. 62 yx 16. 6
43
21
yx
AlgebraQuiz Review 6.2
Slope:
100. 5
32
xy200. 1835 yx
300. )3,1( and )13,5( 400. )5,2( and )4,3(
500. 2
43
95
yx600.
xxy
3
)3(7
Intercepts:
100. 632 yx 200. 1545 yx
300. 783 xy 400. 5
43
21
yx
500. xyyx 5)(2 600. 72
113
21
yx
AlgebraQuiz Review 6.2Slope-Intercept Form:
100. xy 3 200. 1
43
xy
300. 2
31
21
xy400.
)5(543 xy
500. 31
512 yx
600. 232
yx
Standard Form:
100. 5 xy 200. 5
21
xy
300. xy
523
400. 352
yx
500. 103
52
yx600.
)(72)(
21 yxyx
AlgebraPractice Quiz: Linear EquationsName________________________ Period _____
6.2Convert Each to Standard Form and list values for A, B, and C.
1. 6 xy1. A=____B=____C=____
2. xy 72 2. A=____B=____C=____
3. yx 52
21
3. A=____B=____C=____
4. y412
4. A=____B=____C=____
5. 23
y
x
5. A=____B=____C=____
State the y-intercept of each equation below:
6. 4 yx6. y-int. _______
7. 341
xy
7. y-int. _______
8. yx213
8. y-int. _______
9. 352 yx9. y-int. _______
10. yx 510. y-int. _______
AlgebraName________________________ Period _____
Practice Quiz: Linear Equations 6.2Write an equation in slope-intercept form for each:
11. _______________
12. _______________
13. _______________
14. _______________
15. _______________
State the slope for each equation or pair of points:
16. 1052 yx16. m=_______
17. xy 32
17. m=_______
18. )10,9( and )5,1( 18. m=_______
19. )1,3( and )5,4(19. m=_______
20. )12,4( and )12,4(20. m=_______
15
1211
13
14
AlgebraPractice Quiz: Lin. Equations (4) 6.2State the slope of each line graphed below:
11. m=_______
12. m=_______
13. m=_______
14. m=_______
15. m=_______
State the slope for each equation or pair of points:
16. 1052 yx16. m=_______
17. xy 32
17. m=_______
18. )10,9( and )5,1( 18. m=_______
19. )1,3( and )5,4(19. m=_______
20. )12,4( and )12,4(20. m=_______
15
14
13
1211
Name________________________ Period _____
AlgebraSelf-Check: Linear EquationsName________________________ Period _____
Graph each. Remember to extend your lines to the edge of the graphs.Label each graph with a 1, 2, or 3.
1. 92 xy
2. 63 yx
3. 1832 yx
Self-Check: Linear EquationsName________________________ Period _____
6.0Graph each. Remember to extend your lines to the edge of the graphs.Label each graph with a 1, 2, or 3.
4. 84 yx (careful!)
5. 672 yx (think!)
6. 34 yx (think!)
AlgebraSlope and Standard FormPractice: Convert each of the following Standard-Form equations intoSlope-Intercept Form. State the slope of each.
1. 22 yx 2. 1234 yx 3. 852 yx
There is a simple formula that can be used to find the slope of any Stan-dard Form equation. Try to find it be solving Standard Form for Slope-Intercept Form:
CByAx becomes xy
Examples: State the slope of each equation:
1. 432 yx 2. 5 yx 3. 37 yx
Practice: State the slope of each equation:
1. 953 yx 2. 13 yx 3. 352 yx
4. 41138 yx 5. 799 yx 6. 354 yx
Practice: For each of the following, find the slope and one interceptwithout converting. Graph each:
1. 852 yx 2. 63 yx 3. 272 yx
AlgebraSlope and Standard FormName________________________ Period _____
Graph each. Remember to extend your lines to the edge of the graphs.Clearly label each graph with a 1, 2, 3, or 4.
1. 1032 yx
m=
2. 1023 yx
m=
3. 34 yx
m=
4. 272 yx
m=
Graph each. Remember to extend your lines to the edge of the graphs.Clearly label each graph with a 5, 6, 7, or 8.
5. 72 yx
m=
6. 834 yx
m=
7. 2729 yx
m=
8. 2498 yx
m=
AlgebraPoint-Slope Form 6.4Practice: Write an equation for each in Slope-Intercept Form:
We need a new form!POINT-SLOPE FORM
Given any point on the line ),( 11 yxand the slope of the line m
)( 11 xxmyy MEMORIZE THIS!
Examples:Write an equation in Point-Slope Form using the information given.
1. )3,5( 21
m 2. )9,5( 52
m
Practice:Write an equation in Point-Slope Form using the information given.
1. )1,4( 81
m 2. )3,6( 2m
3
2
1
AlgebraPoint-Slope Form 6.4You can write a Point-Slope equation given any two points.
Try it on your own: Write a Point-Slope equation
for the line that passes through )7,1( and )5,2( .
Practice:Write an equation in Point-Slope Form using the information given.
1. )2,6( )3,1( 2. )11,14( )5,6(
Practice:Write an equation in Point-Slope Form for each graph.Use the darkened point.
Practice:Convert each equation you got for the lines above into Standard Form:
1. )2(343 xy 2. )1(
315 xy 3. )7(
322 xy
32
1
AlgebraSlope Mazes 6.1Review:To solve each maze, you must find the nearest point using the slope given beloweach blank.
Example: Start at point A. Which point would come next if the slope were -3?
Practice: Start at point G. Follow the slopes and write the sequence of letters you use.There will not be a word spelled.
_G_ _____ _____ _____ _____ _____ _____ _____
41
101
7 1 0 21
1
Name________________________ Period _____
P
A
D
F
G
C H
Y
ZW
VU
R
N
TS
Q
JK
I
ML
X
B
E
AlgebraSlope Mazes 6.1Challenge:
Practice: Start at point H. You will make a phrase.
_H_ _____ _____ _____ _____ _____ _____
72
6 319
7 72
1
_____ _____ _____ _____ __?__ __L__
316
7 710
74
1 2
_____ _____ _____ _____ Challenge: What are the coordinates
56
21
1 74
of the missing letter? ______
Name________________________ Period _____
P
A
D
F
G
C H
Y
ZW
VU
R
N
TS
Q
JK
I
ML
B
E
AlgebraSelf Check: Four Formulas 6.4Review:Write each of the four formulas listed below in the blank provided:
Slope: _______________________ Slope-Intercept Form: ______________
Standard Form: ________________ Point-Slope Form: _________________
Name________________________ Period _____
Self Check: Four FormulasWrite an equation for each inStandard Form:
1. __________________
2. __________________
3. __________________ 3
2
1
AlgebraReview 6.4Practice:Write an equation for each line graphed below in the form listed.
1. Slope-Intercept:
2. Point-Slope Form:
3. Standard Form:
1
2
3
Practice:Write an equation for each line graphed below in all three forms:
1.
2.
3.
1
2
3
AlgebraReview: Four Formulas 6.4Find the slope between each pair of points:
1. )3,4( )7,5( 2. )1,9( )0,2(
3. )7,6( )7,3( 4. )4,8( )10,8(
5. )1,7( )2,11( 6. )3,2( )3,2(
Write an equation for each pair of points below in Point-Slope Form, then convertit into both Standard and Slope-Intercept Forms:
7. )1,1( )11,6( 8. )3,5( )4,2(
Point-Slope: _____________________ Point-Slope: _____________________
Standard: _______________________ Standard: _______________________
Slope-Intercept: __________________ Slope-Intercept: __________________
9. )2,7( )7,3( 10. )5,2( )15,9(
Point-Slope: _____________________ Point-Slope: _____________________
Standard: _______________________ Standard: _______________________
Slope-Intercept: __________________ Slope-Intercept: __________________
Name________________________ Period _____
AlgebraReview: Four Formulas 6.4Write an equation for each in the form listed:
11. Standard
12. Slope-Intercept
13. Standard
14. Point-Slope
Name both Intercepts for each equation:
15. 4052 yx 16. 1037 yx
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
17. 117 yx 18. 341720 yx
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
19. yx 515 20. 52 xy
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
Name________________________ Period _____
12
11
13
14
AlgebraParallel/Perpendicular Lines 6.5Graph the following linear equations on the SAME GRAPH:
A. 543
xy
B. 1643 yx
C. 134
xy
D. )9(346 xy
The slopes of lines that are parallel are ___________________.
The slopes of lines that are perpendicular are ______________.
Examples:Find the parallel AND perpendicular slopes for each:
1. 21
m 2. )9,5( )6,4( 3. 53 yx
Practice:Find the parallel AND perpendicular slope for each:
1. 3m 2. )7,2( )2,8( 3. 1472 yx
AlgebraParallel/Perpendicular Lines 6.5Examples:Write the equation for each of the following:
1. Parallel to 321
xy through )2,5( in Point-Slope Form:
2. Perpendicular to 32 yx through )7,3( in Standard Form:
Practice:Write the equation for each of the following:
1. Parallel to )3(522 xy through )5,3( in Point-Slope Form:
2. Perpendicular to 53 yx through )4,2( in Standard Form:
3. Perpendicular to )1(53 xy through )8,1( in Slope-Intercept Form:
Practice:
1. Write the Point- Slopeequation for the linepassing through bothpoints to the right.
2. Write the equation for aperpendicular line passingthrough point A in Point-Slope Form.
3. Write the equation for aperpendicular line passingthrough point B inStandard Form.
A
B
AlgebraTry writing an equation and graphing some of the ‘real-life’ problems below:
It is VERY IMPORTANT to Remember: y is the dependent variable, x is theindependent variable, y always depends on x.
Examples: Label your variables, then write an equation for each:
Slope-intercept form:A bear cub weighs 8kg at birth and gains 3/4 kilogram per week.
Point-Slope Form:Each of the Keebler elves can make 9 batches of cookies in 4 hours, and 15
batches in 6 hours. (to begin: write two points, then find the slope)
Slope-intercept form:Expenses are $75 to rent the space and then $15 per guest.
Point-Slope Form:In the same taxi, you went 5 miles for $13, while a 13-mile trip cost $29.
Practice: Write a linear equation for each:
1. Mailing a medium-sized package costs $5 plus $1.50 a pound.
2. A baby weighs 14 pounds at 5 months and 21 pounds at 10 months.Convert this equation to slope intercept form and answer:a. How much did the baby weigh at birth?b. How many pounds did the baby gain each month in its first year?
More Practice: Write an equation for each:
1. A restaurant delivers pizzas for $8.95 each plus a $4 charge for delivery.
2. Express-mailing a 12-pound package costs $13, while it costs $34 to mail a40-pound package express.
3. A Sprint cell-phone plan charges a $0.50 connection fee and then $.05 aminute for each call.
4. With an AT&T cell-phone plan, you pay $.74 for a 7-minute call, and$3.05 for 40 minutes. What is the connection fee for AT&T?
Linear Modeling (word problems) 6.6
AlgebraWrite an equation to represent each situation given below in the form listed.Convert each to the form listed.
1. Mario’s Pizza charges $7 for a medium pizza plus $0.75 per additional topping.
Slope-Intercept Form: (use c for charge and t for toppings)
______________________
Standard Form:
______________________
2. A taxi ride in Boston costs $11 for 2 miles, and $18 for 4 miles.
Point-Slope Form: (use c for cost and m for miles)
______________________
Slope-Intercept Form:
______________________
3. A long distance company charges a $1 connection fee, plus $0.10 a minute.
Slope-Intercept Form: (use m for minutes and c for charge)
______________________
Standard Form:
______________________
4. It costs 85 cents for a 12-ounce beverage, and $1.25 for a 20-ounce beverage:
Point-Slope Form: (Use n for ounces and c for cost)
______________________
Standard Form:
______________________
Word Problems 6.6Name________________________ Period _____
AlgebraWrite an equation to represent each situation given below in the form listed.Convert each to the form listed.
5. A calf weighs 18 lbs when it is 2 months old, and after 8 months weighs 36 lbs.
Point-Slope Form: (use w for weight and m for months)
______________________
Slope-Intercept Form:
______________________
6. Shipping an internet purchase costs $3 plus $0.50 a pound.
Slope-Intercept Form: (use p for pounds and c for charge)
______________________
Standard Form:
______________________
7. An automotive factory makes 17 cars in 5 hours and in 8 hours canmake 29 cars.
Point-Slope Form: (use h for hours and c for cars produced)
______________________
Slope-Intercept Form:
______________________
8. A rental car charges $29 to rent the car plus $45 a day:
Slope Intercept Form: (Use d for days and r for the rental fee)
______________________
Standard Form:
______________________
Word Problems 6.6Name________________________ Period _____
AlgebraOther ShortcutsYou can write an equation in slope-intercept form given two pointson the line without using point-slope form.
Example:Write the equation of the line passing through (-5, -5) and (5, 1) in slope-inter-cept form.
Method 1: Use point-slope form and convert.Method 2: Find the slope, then solve for b in slope-intercept form.
Practice:Write a slope-intercept form equation for each pair of points.(Practice method 2.)
1. (5, 14) (-1, -4) 2. (6, -4) (-2, -8)
3. (-8, -4) (4, 5) 4. (-2, 9) (11, -5)
You can write an equation in Standard Form given two points on theline without using point-slope form.
Example:Write the equation of the line passing through (-5, -5) and (5, 1) in StandardForm.
Method 1: Use point-slope form and convert.Method 2: Find the slope, use it for A and B, then solve for C.
Practice:Write a Standard Form equation for each pair of points.(Practice method 2.)
1. (5, 14) (-1, -4) 2. (6, -4) (-2, -8)
3. (-8, -4) (4, 5) 4. (-2, 9) (11, -5)
AlgebraWrite an equation for each given the information listed in the form listed.
1. Write an equation in slope-intercept form for the line with slope 2/3 which passes throughthe point (6, -2).
slope-intercept form: _____________________
2. Write an equation in Standard form for the line with slope -4/5 which passes through thepoint (3, -5).
standard form: _____________________
3. Write an equation in slope-intercept form for the line which passes through the points(2, 5) and (6, 3).
slope-intercept form: _____________________
4. Write an equation in Standard form for the line which passes through (4, 6) and (2, -1).
standard form: _____________________
5. What is the standard form equation of the line parallel to 2x-7y=5 which passes throughthe point (3, -2).
standard form: _____________________
6. Write the slope-intercept form of the line perpendicular to y=3x+7 which passes throughthe point (6, -5).
slope-intercept form: _____________________
7. Write the standard form of the equation of a line passing through (7, -2) and (2, -3).
standard form: _____________________
8. Write a point-slope equation to represent the line that passes through the point (6, -2) andis perpendicular to teh line which passes through (7, 8) and (-2, 5).
slope-intercept form: _____________________
Using ShortcutsName________________________ Period _____
AlgebraTest Review 6.2Slope-Intercept Form:
100. Convert )8(
432 xy
to Slope-Intercept Form.
200. Through )2,4( and )4,3( in Slope-Intercept Form.
300. Perpendicular to 59415 yx through )11,2( in Slope-InterceptForm.
Slope:
100. Find the slope between )2,4( and )4,3( .
200. Find the perpendicular slope to the graph of: 5y
300. Find the slope of a line parallel to: 554
32
xy
Word Problems:
100. A tow truck charges $25 to pick you up plus $3 a mile for the tow.(c=charge, m=miles)
200. Michael made 5 pancakes in 30 minutes, and 10 pancakes in 40 minutes.(p=pancakes, m=minutes)
300. A phone company charges $0.50 the first minute and $0.15 for everyminute after that. (c=charge, m=minutes)
AlgebraTest Review 6.2Point-Slope Form:
100. Through )5,1( and )4,3( in Point-Slope Form.
200. Parallel to 43 yx through )8,2( in Point-Slope Form.
300. Perpendicular to 5
21
32
yx through )7,2( in Point-Slope Form.
Standard Form:
100. Convert )4(213 xy to Standard Form.
200. Parallel to 42 yx through )1,1( in Standard Form.
300. Write an equation in Standard Form for the line whose x-intercept is -2and whose y-intercept is 13.
AlgebraState the x and y-intercepts of each:
1. 14 xy1. x-int. _______ y-int. ______
2. 82 yx2. x-int. _______ y-int. ______
3. 521
yx
3. x-int. _______ y-int. ______
Write an equation for each line graphed below in the form listed:
4. Standard Form:
___________________________
5. Point-Slope Form:
___________________________
6. Slope-Intercept Form:
___________________________
Find the slope for each equation or pair of points:
7. xy 43 7. slope: ________________
8. )1,2( )3,5( 8. slope: ________________
9. 7y9. slope: ________________
Practice Test: Linear Equations 6.6Name________________________ Period _____
4
5
6
AlgebraFind the slope of each line:
10. slope: ________________
11. slope: ________________
12. slope: ________________
Write a Point-Slope equation for each, then convert to the form listed:
13-14. )5,3( )2,4( 13. Point-Slope Form: ______________________
14. Standard Form: ______________________
15-16. )9,4( )6,5(15. Point-Slope Form: ______________________
16. Slope-Intercept Form: ______________________
Write an equation in Standard Form of a line parallel to the equation below whichpasses through the point given.
17. 2032 yx )3,8( 17. Standard Form:__________________
Write an equation in Slope-Intercept Form of a line perpendicular to the equationbelow which passes through the point given.
18. 155 yx )2,3(18. Slope-Intercept Form:__________________
Practice Test: Linear Equations 6.6Name________________________ Period _____
10
11
12
AlgebraSemester Review S1Part 1. Solving Equations
Solve for x.
1. 1947 xx 2. 712 x
3. 5
826
x4. 1652 x (graph your answer)
Part 2. Proportional Reasoning
Solve:
1. Find the missing length x 2. What percent is 9 of 25?in the similar trianglesbelow:
3. What number increased by 15% is 92?
Part 3. Linear Equations
Solve:
1. State the slope between: )2,4)(9,1(
2. State the x and y intercepts: 2052 yx
3. Write an equation for a line perpendicular to 32 xy
Which passes through the point )1,6( in Standard Form.
5.4cm
9.9cm
3.3cm
x cm
8.3cm
AlgebraSemester Review S1Part 4. Exponents
Simplify
1. )5(3 23 yxyx 2.
34 )2( ba
3. 5
35 )2(xx
4.
2
75
32
baba
Part 5. Systems of Equations
Solve Each Using Substitution or Elimination:
1. Solve for x and y: 52 xy
732 yx
2. Solve for x and y: 1623 yx
232 yx
3. Jamie has a cup full of quarters and dimes. She has a total of 17 coins, for atotal of $3.05. How many of each coin does she have?
4. Eddie is mixing two types of candy. Runts cost $5 a pound, and caramelscost $4 a pound. If he mixes 8 pounds worth $4.25 a pound, how muchof each candy did he use?
AlgebraPractice Semester Exam 1 S1Name________________________ Period _____
Solve for x:
1. cbxa )(1. ______
2. 52
74
x
2. ______
3. 35 x3. ______ or ______
4. 39523 x4.______ or ______
Solve:
5. What number decreased by 30% is 28?
5. ______
6. The price of a stereo was $229.99, and you paid $250.69 at the register. What percent
was the sales tax?
6.______
What are the x and y-intercepts for the line described below?
7. Slope:43
Through the point: )5,2(7. x-int.______ y-int.______
Write an equation in slope-intercept form for a line passing through:
8. )1,3( and )4,6( 8. ______
AlgebraPractice Semester Exam 1 SName________________________ Period _____
Find the point where the lines below intersect:
9. 53 xy23 yx
9.____________
Solve each system of equations below:
10. 1632 yx184 yx
10.____________
11. 203 yx6 yx
11.____________
Solve:
12. A Ligon student is raising money for a fund raiser. She sells Blow Pops for $0.25
each and candy bars for $0.75 each. She has sold 21 items for a total of $14.25.
How many of each has she sold?
12. B(Blow Pops)=_______ C(Candy bars)=_______
Simplify:
13. 2352 )2(3 yxyx
13. ______
14. 32
17
5)(3
yxyxxy
14. ______
15.
3
2
42
aa
15. ______
Pledge: Write-out and sign.
AlgebraPractice Semester Exam 2 S2Name________________________ Period _____
Solve for x:
1. bd
cxa
)(
1. ______
2. 65
3)7(2
x
2. ______
3. 12752 x3. ______
4. 18533 x4.______ or ______
Solve:
5. What number increased by 30% is 52?
5. ______
6. A stereo is on sale for 10% off, and now costs just $143.10. What was the original price
of the stereo before the sale?
6.______
What is the slope of the line below?
7. 723 yx7. m=______
Find the x and y-intercepts for a line passing through the points below:
8. )4,6)(1,3( 8. x-int.______ y-int.______
AlgebraPractice Semester Exam 2 S2Name________________________ Period _____
Write an equation for a line perpendicular to the line below andpassing through the point given in Standard Form:
9. Perpendicular to 532 yx through )2,4(9. Standard Form_________________
Solve each system of equations below:
10. 532 yx114 yx
10.____________
11. 83 yx6 yx
11.____________
Solve:
12. Juan buys six Blow Pops and three candy bars for $4.05. The next day, he buys
seven Blow Pops and two candy bars for $3.75. How much will he pay for five Blow
Pops and four candy bars?
12. 5 Blow Pops and 4 candy bars =_______
Simplify:
13. 33 )2( xy13. ______
14. yxyxyx
2
272
12)(3
14. ______
15.
3
5
32
xyx
15. ______
AlgebraPractice Semester Exam 3 S3Name________________________ Period _____
Solve for x:
1. cdxba )(1. ______
2. 7
3253
xx
2. ______
3. 52
3 x
3. ______
4. 1712 x4.______ or ______
Solve:
5. 14% more than what number is 9.12?
5. ______
6. Amanda has 25% more money than Claire. If Amanda has $123, how much money
does Claire have?
6.______
What is the slope of a line parallel to the line of the equation below?
7. 32 yx7. m=______
The graph of the line passing through the two points below is shifted UP three units. What is the y-intercept of the resulting graph?
8. )12,3)(2,4( 8. y-int.______
AlgebraPractice Semester Exam 3 S3Name________________________ Period _____
Write an equation for a line perpendicular to the line below and
passing through the point given in Slope-Intercept Form:
9. Perpendicular to 1856 yx through )11,12(9. Slope-Intercept Form_________________
Solve each system of equations below:
10. 8 xy123 yx
10.____________
11. 352 yx1052 yx
11.____________
Solve:
12. A test consists of true/false questions and fill-in-the-blank questions. There are 24
questions on the test. If the true/false questions are worth three points each and
the fill-in-the-blank questions are worth seven points each for a total of 100 points,
how many of the questions are fill-in-the blank questions?
12. fill-in-the-blank questions =_______
Simplify:
13. 42 )2(3 yy13. ___________
14. yxyxxy
3
3232 )(
14. ___________
15.
3
7
4
105
xx
15. ___________
AlgebraReview: Four Formulas ReName________________________ Period _____
Find the slope between each pair of points:
1. )3,4( )7,5( 2. )1,9( )0,2(
3. )7,6( )7,3( 4. )4,8( )10,8(
Write an equation for each pair of points below in Point-Slope Form, then convertit into both Standard and Slope-Intercept Forms:
5. )1,1( )11,6( 6. )3,5( )4,2(
Point-Slope: _____________________ Point-Slope: _____________________
Standard: _______________________ Standard: _______________________
Slope-Intercept: __________________ Slope-Intercept: __________________
Write an equation parallel to the Write an equation perpendicular to thegiven equation and through the given equation and through thepoint given: point given:
7. 832 yx )7,3( 8. 75 xy )7,5(
Point-Slope: _____________________ Point-Slope: _____________________
Standard: _______________________ Standard: _______________________
Slope-Intercept: __________________ Slope-Intercept: __________________
AlgebraReview: Four Formulas ReWrite an equation for each in the form listed:
9. Standard
10. Slope-Intercept
11. Standard
12. Point-Slope
Name both Intercepts for each equation:
13. 4052 yx 14. 1037 yx
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
\
15. yx 515 16. 52 xy
x-int.: _______ x-int.: _______
y-int. ________ y-int. ________
Name________________________ Period _____
12
11
13
14
AlgebraPractice Semester Exam 4 S4Name________________________ Period _____
Solve:
1. A pair of sunglasses costs $29.95 but is on sale for 15% off. After a 7.5% sales tax, how much will the sun
glasses cost? (to the cent)
1. ______
2. Ryan can run a quarter mile 15% faster than his sister Ally. If Ryan can run a quarter
mile in 68 seconds, how many seconds does it take Ally?
2.______
3. Write an equation in Standard Form for a line with a slope of 97
and a y-intercept of -5:
3. _______________________
The line which passes through the two points below is shifted to the right 5 units.What is the new x-intercept of the graph?
4. )20,3)(6,4( 4. x-int.______
Solve for x:
5. cxba )(5. __________
6. 4
1532
x
6. __________
7. 254 x7. __________
Solve for x and graph your solution on the line provided:
8. 172 x8.______________ ______
AlgebraPractice Semester Exam 4 S4Name________________________ Period _____
Write an equation for a line parallel to the line below and
passing through the point given in Slope-Intercept Form:
9. Parallel to 15 yx through )10,2( 9. Slope-Intercept Form_________________
Simplify:
10. 32 )2(5 xyxy
10. ___________
11. 3
232
10)5(
baba
11. ___________
12.
2
7
2
820
xyx
12. ___________
Solve:
13. Marianna makes $6.50 an hour as a lifeguard and $7.50 an hour as a cashier
during the summer. In one week she works for 19 hours and makes $132.50. How
many hours did she work as a cashier?
13. hours as a cashier =_______
Solve each system of equations below:
14. 115 yx20112 yx
14.____________
15. 1432 yx126 yx
15.____________
Bonus: Circle the equation that does NOT represent a function:
a. 3 xy b. xy 3 c. 32 yx d. 23xy
AlgebraGraphing a System of Equations 7.1Given two equations, the solution is the point that satisfies both.
Graphing is the first way we will learn to solve a system of equations.
Example:Find the solution to thefollowing system ofequations by graphingthem.
532
xy
221
xy
Practice:Graph each and find thesolution to each pair ofequations:1. a & b2. b & c3. a & c
a. 431
xy
b. 12 xy
c. 4 yx
AlgebraGraphing a System of Equations 7.1Practice:Use a graph to determine the solution to each system of equations:
1. 23 xy &
743
xy
2. 4 yx &
621
xy
More Practice: Solve.
1. )5(322 xy &
)4(211 xy
2. 2052 yx &
11 yx
AlgebraGraphing Inequalities 7.6Graphing Inequalities in Slope-Intercept FormWorks the same as graphing equations except:Dash the line for < or >Shade above if y >Shade below if y <
Examples:
1. 52 xy
2. 631
xy
3. 421
xy
4. 15 xy
Practice:
1. 64 xy
2. 351
xy
AlgebraSystems of Inequalities 7.6Graphing a System of InequalitiesGraph and lightly shade each inequality.Darken the area of overlap.Test a point in the darkened area to check your graph.
Examples:
1. 62 xy& 5 xy
2. 143
xy
& 102 xy
Practice:Graph each systemof inequalities.
1. 352
xy
& 523
xy
2. 453 xy
& 2 yx
AlgebraYou can make comparisons by graphing equations.
Practice:Compare three towing companies by writing an equation and graphing the
charge of a tow based on the number of miles you need to be taken.
Auto Shop towing:$15 to come pick you up, $.50 a mile for the tow.
Benny’s wrecker service:$10 to come pick you up, $.75 a mile for the tow.
Cary Automotive:6 miles cost $10, 12 miles costs $19 (begin in point-slope, change to slope-
intercept form)
Answer:After how many miles are A and B the same price?After how many miles are B and C the same price?After how many miles are A and C the same price?
For what mileage is A the best deal?For what mileage is B the best deal?For what mileage is C the best deal?
Systems of Equations 7.6
AlgebraGraph each pair of equations below toanswer the questions that follow:
1. The Yellow Cab Company charges just $0.25 a mile, but it costs $5 to get inthe cab. Express Cab charges no fee to get in the cab, but $1.50 a milefor the ride.
a. If you are going 7 miles, which cab company should you call?
b. If you are going 3 miles, which company should you call?
c. For what length of drive is the cost equal?
2. Ashley and Emma are reading the same article. Ashley is on page 1 of thearticle, but she can read a page every minute. Emma is already on page5, but reads a page every three minutes.
a. What page is Ashley on after 5 minutes?
b. What equation could be used to represent the amount Emma has read?
c. How many minutes does it take before Ashley and Emma have read thesame amount?
3. David and James are at the Famous Nathan’s Hot Dog Eating Champion-ships of the world in New York on the 4th of July. David was late starting,so James already had 6 hot dogs before David started eating. Form thenon, James ate a hot dog every two minutes while David stuffed 1¼ hotdogs a minute.
a. What equations could you use to compare David and James’ hot dogeating?
b. Between David and James, who would win the contest if it lasts 12 minutes?
c. How many minutes does it take David to catch up with James?
Systems of Equations 7.6
AlgebraSanta’s Elves 7.6Name________________________ Period _____
Santa’s elves are hard at work, and December is of course their busiest month. Four ofSanta’s elves are competing for the ‘elf of the year’ award, which is given to the elf whohas the most completed toys by Christmas Eve.
Write an equation to represent each Elf below. Graph the equation for each elf on theback of this sheet and answer the questions that follow.
Alex ‘slow and steady’ McElfHad 13 toys made to begin the month, and makes one new toy every four days.
Equation: _______________Bob ‘the procrastinator’ ElfingtonAfter 3 days had only four toys made, but after nine days had ten toys made.
Equation: Point-Slope_______________ Slope-Intercept _______________
Cramden ‘up all night’ ElfmanWorks 24 hours, all day and night, and manages to make one toy every 36 hours. He began themonth in third place with 8 toys made. (Think about the slope!)
Equation: _______________
Duke ‘the maniac’ S’elfishStarted the month with the most toys made (17), but the evil elf has been smuggling them out of theshop to sell on Ebay at a rate of one every three days. (He has a negative slope).
Equation: _______________1. On which days is Cramden in the lead, or tied for the lead?
________________________
2. On what days is Alex in the lead or tied for the lead?
________________________
3. When are Bob and Cramden tied?
________________________
4. How many days does it take Bob to get out of last place?
________________________
5. Think carefully: On what day does Duke lose the lead?
________________________
6. Which elf should win the award (on the 24th)?
________________________
AlgebraGraphing Inequalities 7.6Graphing Inequalities in other forms:If an equation is hard to convert, or has a y-intercept that is not integral,Graph in Standard or Point-Slope Form and pick a point to test which side toshade.
Examples:Point-Slope Form
1. )1(316 xy
2. )9(237 xy
Practice:Graph the following systemof inequalities.
1. )9(216 xy
& )10(32 xy
AlgebraReview: Graphing Equations 7.6Name________________________ Period _____
Solve each system of equations below by graphing:(two problems per graph)
1. 552 xy solution: ________
& 32 xy
2. 521
xy solution: ________
& 303 yx
3. 73 yx solution: ________
& 3 yx
4. 291
xy solution: ________
& 1532 yx
5. xy 2 solution: ________
& )8(316 xy
6. )5(433 xy solution: ________
& )8(527 xy
AlgebraReview: Graphing Inequalities 7.6Name________________________ Period _____
Graph each system of ineqalities NEATLY.
7. 832
xy 8. 221
xy
& 32 xy & 2052 yx
9. 552 xy 10. )8(32 xy
& )9(315 xy & )10(
314 xy
AlgebraPractice Quiz: Graphing Systems 7.6Name________________________ Period _____
Solve each system of equations below by graphing:(two problems per graph)
1. 431
xy solution: ________
& 102 xy
2. 621
xy solution: ________
& 352
xy
3. 13 xy solution: ________
& )7(321 xy
4. 1834 yx solution: ________
& 5 yx
Inequalities: Graph and shade.
5. 123
xy
& 142 yx
AlgebraPractice Quiz: Graphing Systems 7.6Name________________________ Period _____
Solve each with a graph:
6-7. One phone company charges a $1 connection fee and $0.50 a minute for calls to Australia. Asecond company charges a connection fee of $5, but only charges $0.25 a minute.
6. How long is a phone call that costs the same
with both companies? _______
7. How much does it cost? _______
8-9. Jared can make one Christmas ornament a minute, and he already has five made. Marissacan make three ornaments every two minutes, but does not have any made.
8. If they both start working at the same time,
how many minutes will it take for Marissa
and Jared to have the same number of
ornaments made? _______
9. How many ornaments does each have
made when they are tied? _______
Pledge:
AlgebraQuiz: Graphing Systems 7.6Name________________________ Period _____
Solve each system of equations below by graphing:(two problems per graph)
1. 631
xy solution: ________
& 2 yx
2. 852
xy solution: ________
& yx21
3. 113 xy solution: ________
& )3(345 xy
4. 9 yx solution: ________
& xy 45
Inequalities: Graph and shade.
5. 621
xy
& 623 yx
AlgebraQuiz: Graphing Systems 7.6Name________________________ Period _____
Solve each with a graph:
6-7. Candy is sold by the ounce at two stands at the mall. One stand charges $1.50 per ounce in afree bag. A second stand charges $6 for a jar that you can fill for $0.75 per ounce.
6. How many ounces must be bought for
the cost of the bag of candy to equal the cost
of the jar? _______
7. What is the cost when they are
equal? _______
8-9. Ken and Kayla are reading a book. Ken is already on page 8 and reads a page every twominutes. Kayla has just started reading and can finish a page in just 40 seconds.
8. How many minutes does it take for Kayla
to reach the same page as Ken? _______
9. What page are they on when Kayla and
Ken begin the same page? _______
Pledge:
AlgebraSubstitution 7.2Review:Solve each of the following equations for y using the given value for x.
1. 23 xy for 7x
2. 352
xy for 10x
3. 92 xy for 3 yx
Substitution:Mehtod 1: GraphingMethod 2: Substitution
To solve a system of equations using substitution:Solve one equation for x (or y).Substitute this value into the other equation and solve for y (or x).
Ex. 93 xy and 7 xy
Harder Example 6327 xy and 152 yx
Practice:Solve each system using substition.
1. 53 xy 2. 52 yx9 yx 6 xy
3. 223
xy 4. 135 yx
12 xy 12 yx
AlgebraSubstitution 7.2Review:Solve each system below using substitution.
1. 321
xy for 62 yx
2. 732
xy for 1232 yx
When solving a system using substitution, you sometimes arrive at a ‘dead end’.
Examples of ‘No Solution’: 3=2 or 5=0If you get to x=3x, this does NOT mean there is no solution. What value worksin this case for x?
Examples of ‘Infinite Solutions’ (Identities): 3=3 or 2x=2x or x-3=x-3
Practice:Solve each system using substition. Write No Solution or Infinite Solutionswhere applicable.
1. 5 xy 2. 83 yx9 yx 83 xy
3. 221
xy 4. 124 yx
42 yx 43 yx
AlgebraElimination 7.3Review:Solve each of the following equations using Substitution:
1. 52 xy 2. 2653 yx52 yx 1254 yx
Elimination:Method 1: GraphingMethod 2: SubstitutionMethod 3: Elimination
To solve a system of equations using elimination:Add the two equations to eliminate a variable (x or y).Adjust the equations with multiplication before adding them if necessary.
Ex. 1254 yx and
2653 yx
Harder Example: 1032 yx and
3165 yx
Practice:Solve each system using elimination.
1. 92 yx 2. 1734 yx163 yx 1152 yx
Practice:Solve each system using elimination.
1. 33 yx 2. 2353 yx3023 yx 135 yx
AlgebraSubstitution and Elimination 7.3Review:Solve each of the following using substitution or elimination:
1. 23 xy 2. 732 yx823 yx 534 yx
Use Substitution when at least one variable has a coefficient of 1 (or -1).Use Elimination when variables share the same coefficient.Both will always work, if neither of the above is true, use whichever method youare more comfortable with.
Examples:Substitution or Elimination? (DO NOT SOLVE)
1. 53 xy 2. 1125 yx 3. 313 yx3 yx 322 yx 53 xy
Now, solve them.
1. 53 xy 2. 1125 yx 3. 313 yx3 yx 322 yx 53 xy
Use Substitution or Elimination to solve the following.
1. 82 xy 2. 835 yx 3. 125 yx032 yx yx 224 11 yx
AlgebraSubstitution and Elimination 7.6Name________________________ Period _____
Substitution and Elimination:Solve each using substitution or elimination.
1. 113 xy 2. 5 yx32 yx 3 yx
3. 112 yx 4. 2 yx3 yx 22 xy
5. 53 xy 6. 23 yx7 xy 102 yx
7. 12 xy 8. 23 xyxy 4 yx 32
9. 1132 yx 10. 52 xy2 yx 1042 yx
11. 532 yx 12. 12 yx112 yx 46 yx
AlgebraSubstitution and Elimination 7.6Name________________________ Period _____
Substitution and Elimination:Solve each using substitution or elimination.
13. 243 yx 14. 33 xy1244 yx 1223 yx
15. 2432 yx 16. xy 4186 yx 72 yx
17. 43 yx 18. 1123 yx
562 yx 421
yx
19. 5.02.03.0 yx 20. yx 27 152 yx 94 yx
Algebra 7.6Name________________________ Period _____
Solve each system of equations below by graphing:(two problems per graph)
1. 531
xy solution: ________
& 92 xy
2. 621
xy solution: ________
& 732
xy
3. 3 xy solution: ________
& xy52
4. )1(419 xy solution: ________
& 22 yx
Inequalities: Graph and shade.
5. 631
xy
& 13 xy
Practice Quiz: Systems of Equations
AlgebraPractice Quiz: Systems of Equations 7.6Name________________________ Period _____
Solve each system of equations below by using substitution or elimination.
6. 22 yx solution: 6. __________
& 1823 yx
7. 13 yx solution: 7. __________
& 452 yx
8. 432
xy solution: 8. __________
& 1532 yx
9. 1 xy solution: 9. __________
& 1 yx
10. 434 yx solution: 10. __________
& 1152 yx
11. 6 xy solution: 11. __________
& 244 yx
AlgebraWord Problems: Systems 7.3Word problems:1. Find and label the two variables.2. Write and solve a system of equations.
Example:Tammy works two jobs. As a clerk she earns $7 an hour. As a receptionist she
makes $9 an hour. One week she worked 24 hours and earned $200.How many hours did she work at each job that week?
What are the two VARIABLES?
What equations could compare these two variables?
hint: Money Equation: _____________
Hours Equation: ______________
Solve using elimination OR substitution.
Practice:Write a system of equations and solve:
1. Alyssa scored 54 points in her basketball game. If she made 24 shots, howmany of her shots were 2-pointers, and how many were 3-pointers?
2. Brian sold fruit at his stand. Apples cost $.40 and pears cost $.50 each. Inan afternoon he sold 52 pieces of fruit and made $24. How many of eachdid he sell.
3. Melinda needed to mail a package. She used $.02 stamps and $.10 stampsto mail the package. If she used 15 stamps worth $.78, how many ofeach type of stamp did she use?
20 apples, 32 pears. 9 $.02, 6 $.10
AlgebraWord Problems: Systems 7.6Name________________________ Period _____
Solve each using a system of equations.1. A test contains 35 questions worth a total of 100 points. There are seven-point questions and two-
point questions. How many two-point questions are there? How many seven-point questions?
Equations: x + y = 35 2-pts: _____
__________________ 7-pts: _____show work below!
2. The math club and the science club bought supplies for a retirement home. The math club boughtsix cases of juice and one case of bottled water for $135. The science club bought four casesof juice and two cases of bottled water for $110. How much does a case of juice cost? Howmuch for a case of water?
Equations: 6j + 1b = 135 Juice: _____
_________________ Water: _____show work below!
3. In a parking lot there are motorcycles and cars. You count 98 wheels, and your friend counts 30vehicles. How many cars are there? How many motorcycles?
Equations: m + c = 30 Cars: _____
__________________ Motorcycles: _____show work below!
AlgebraWord Problems: Systems 7.6Name________________________ Period _____
Solve each using a system of equations.4. John sells hamburgers ($3) and cheeseburgers ($3.50). One afternoon he sells a total of 24
burgers for $79. How many of these were hamburgers, and how many were cheeseburgers?
Equations: h + c = 24 Hamburgers: _____
__________________ Cheeseburgers: _____show work below!
5. James paddles upstream in a canoe at 2mph (relative to the shore), and when he paddles down-stream, he goes 9mph. Find the speed of the current (c) and the speed James can paddle instill water (p).
Equations: p + c = 9 Paddle speed: _____
__________________ Current speed: _____show work below!
6. Lisa buys sports supplies for the gym. On Monday, she buys four basketballs and three soccerballs for $85.50. On Tuesday she returns to the store and buys three basketballs and fivesoccer balls for $115. How much do soccer balls cost? How much for basketballs?
Equations: __________________ Soccer balls: _____
__________________ Basketballs: _____show work below!
AlgebraWord Problems: SystemsWord Problems Practice: Money problems.Write a system of equations and solve:
1. Anna has a pocket of dimes and quarters. If she has 10 coins worth $1.45,how many of her coins are quarters?
2. Popsicles cost $0.80, and ice-cream cups cost $0.65. If you purchased 9items for $6.15, how many of the items were popsicles.
Word Problems Practice: Sum/DifferenceWrite a system of equations and solve:
1. The sum of two integers is 19 and their difference is 10. What is the smallerof the two integers?
2. If I add Mark’s age to Tammy’s age, I get 39. If I subtract Mark’s age fromTammy’s age, I get negative 7. What will I get if I multiply Mark’s age byTammy’s?
Word Problems PracticeWrite a system of equations and solve:
1. Mr. Batterson ordered pizzas for the team. Medium pizzas have 8 slices andlarge pizzas have 10. If there are 13 pizzas and 108 slices, how manylarge pizza slices are there?
2. At a toy store, the children’s department has bicycles and tricycles. Thereare 50 total, and 111 wheels. How many bicycles are there?
Word Problems Practice: TimeWrite a system of equations and solve:
1. In five years, Kate will be twice as old as Joey. Right now, Kate is 11 yearsolder than Joey. How old is Joey right now?
2. A bucket is full of red marbles and white marbles. There are twice as manywhite marbles as red ones. If I add seven white marbles, there will bethree times as many white marbles as red ones. How many marbles werein the bucket before the white marbles were added?
AlgebraWord Problems: SystemsName________________________ Period _____
Solve each using a system of equations.
1. A farm has chickens and cows. You ask the farmer how many chickens he has, and how manycows he has. The farmer tells you he has 28 healthy animals, and they have a total of 64legs. How many of his animals are cows?
1. _______
2. Andrew has a collection of soda bottles. Some of them are 12-ounce bottles, and others are 16-ounce bottles. If the collection contains 20 bottles which hold a combined 300 ounces, howmany of the soda bottles are 12-ounce bottles?
2. _______
3. Jack and Cameron are playing a game of paper football. By their rules, you can score a 5-pointtouchdown or a 7-point touchdown. In the game, there have been 13 touchdowns scored fora total of 71 points. How many of these touchdowns were 7-point touchdowns?
3. _______
4. Abbi has $400 in $5 bills and $20 bills. If she has 38 bills, how many of them are $20 bills?
4. _______
5. The sum of two numbers is 40 and their difference is 6.5, what is their product?
5. _______
6. This year, Jake is 5 years older than his sister. Three years ago, Jake was twice his sister’s age.How old is Jake’s sister now?
6. _______
AlgebraWord Problems: Systems 7.3Using PercentsReview: If you have 15 quarts of drink that is 20% Sprite, how many quarts of
Sprite are in the drink?
Example:John is making punch. How many cups of 50% juice should he add to a drink
that contains 10% juice if he wants to make 15 cups of punch containing20% juice? (how many cups of each drink)
x=50% juice y=10% juice
20 yx Total Drink (cups).
)15(2.01.05.0 yx Juice (cups).
Solve using substitution or elimination.
Practice:Write a system of equations and solve:
1. You combine a 10% saltwater mixture with a 40% saltwater mixture to cre-ate 6 gallons of a 30% saltwater solution. How many gallons of eachmixture did you use?
2. Margaret is making fruit punch. She has juice drink that contains 25%orange juice. How much pure orange juice will she need to combine withthe drink to make 17 quarts of a drink that is 60% orange juice?
3. How much of a 90% solution of acid should be added to a 60% acidsolution to create a 5-liter solution that contains 70% acid?
4. Planters is making a new mixture combining Peanuts and Cashews.Cashews cost $7 a pound and Peanuts are $4 a pound. How manypounds of each should be added to make a ten pound mixture thatsells for $4.20 a pound?
Extra: The sum of the digits in a two-digit number is 11. If the digits are re-versed, the number is 27 less than the original. Find the number.
AlgebraWord Problems: Systems 7.6Name________________________ Period _____
Solve each using a system of equations.
1. How much of a 15% vinegar solution should be added to a 35% vinegar solution to make 12 litersof a 20% vinegar solution?
Equations: ________________________ (x) 15% _____
________________________ (y) 35% _____show work below!
2. How many gallons of paint with 40% blue pigment should be added to paint that contains pure(100%) blue pigment to create 20 gallons of a paint that contains 85% blue pigment?
Equations: __________________ (x) 40% _____
__________________ (y) Pure _____show work below!
3. You are taxed at a rate of 5% for all online purchases and 8.5% for all in-store purchases. If youpay a total of 40$ in taxes in addition to spending $500 on purchases (pre-tax), how muchmoney did you spend online, and how much was spent in the store? (before tax, to the cent)
Equations: __________________ (x) online: _____
__________________ (y) in-store: _____show work below!
)12(20.35.015.0 yx
12 yx
40085.005.0 yx
AlgebraWord Problems: Systems 7.6Name________________________ Period _____
Solve each using a system of equations.
4. You have a dish full of nickels and quarters. If there are 16 coins together worth $2.20, how manyof each coin do you have?
Equations: __________________ nickels: _____
__________________ quarters: _____show work below!
5. Two men ask you to guess their ages based on the following clues:The sum of their ages is 76. One of the men is 16 years older than twice the age of theother.
Equations: _________________ (x) 1st man: _____
_________________ (y) 2nd man: _____show work below!
6. When the digits of a two-digit number are switched, the resulting number is 18 less than the origi-nal. If the sum of the digits in the number is 12, find both numbers (show work as a systemof equations, do not use guess-and-check)
hint: Using x as the tens digit, y as the ones digit. 10x+y is the original number, 10y+x is the numberafter the digits are switched.
Equations: __________________ Bigger #: _____
__________________ Smaller #: _____show work below!
16 qn
12 yx
76 yx
AlgebraGraphing Using the TI-83 7+Name________________________ Period _____
Start:1. Turn on your calculator and clear the memory.
Hit 2nd then hit the + symbol and follow the menus to RESET all RAM.(This varies by calculator)
2. Darken or lighten the screen as necessary by hitting 2nd and using the up arrow/down arrow.
Now lets graph some lines.
Find the graph button just below the screen. Push it.
Touch the arow keys. A cursor should appear. You can move it around the screen.
To graph an equation, you need to enter it into your calculator.
y= : Hit the y= button at the top left.This is where you will enter equations to be graphed.Enter the three equations written on the board and hit GRAPH again (I will help explain entering the
equations. Write all three below).
Y1=____________________ Y2=____________________ Y3=____________________
Can you tell which graph is which? Sketch and label the three equations onto the graph below.
TRACE : Hit the trace button.Use the left and right arrows to trace along one of the lines.Use the up and down arrows to switch between lines. The equations should show at the top of the
screen as you switch between lines.Trace until you reach an intersection between lines Y1 and Y2. Can you find the exact point of inter-
section?
CALC : Above the trace button, you will find the word CALC. Hit 2nd then TRACE to get to the CALCmenu.
Choose 5: IntersectFollowing the prompts at the bottom of the screen, select lines Y1 (ENTER) and Y2 (ENTER), then
move the cursor close to the intersection point when it asks for a guess and hit ENTER again.(you do not really need to get that close). Where do lines Y1 and Y2 intersect?
If you mess up the graph, hitZOOM then 6: Standard toget back to the regular graphsetup. We will learn more aboutZOOMing later.
TRACE
AlgebraGraphing Using the TI-83 7+Name________________________ Period _____
Practice:
Graph the following equations and sketch an approximation of the graph from your screen.
1121 xY
8.15.2 xY
456.456.3 xY
012.3091.4 xY
Label the lines you drew Y1, Y2, Y3, and Y4 on your sketch above.
Using the CALC function, find the point of intersection for each system of equations listed below:Round to the thousandth.
1. Y1 and Y2: _________________
2. Y1 and Y3: _________________
3. Y1 and Y4: _________________
4. Y2 and Y3: _________________
5. Y2 and Y4: _________________
6. Y3 and Y4: _________________
Answer:
7. How would you use the calculator to graph an equation that is in Standard Form?
__________________________________________________________________________________
8. Try to graph the following equation: 2772 xy Explain what happened and why..
__________________________________________________________________________________
AlgebraWord Problems: Systems of Ineq. 7.3You can solve a system of Inequalities by graphing word problems.
Example:
1. For a fund raiser, you must raise atleast $30 by selling cookies for $2 abox, and doughnuts for $5 a box.You must sell more than 10 boxes.Graph a system of inequalities toshow all the ways you can do this.
c=cookies d=doughnuts
Inequalities: _______________
_______________
2. Ryan works two jobs. He makes $6an hour working with his dad and$14 an hour mowing lawns. In oneweek, he needs to make at least $84and he only has time to work for amaximum of 10 hours. Graph twoinequalities which show all the wayshe can do this.
d=hours for dad m=mowing hours
Inequalities: _______________
_______________
AlgebraGraphing Inequalities 7.6Name________________________ Period _____
Graph the Following Inequalities:Solve for y if necessary (Slope-Intercept Form).Use a solid or dashed line.Shade the appropriate side.
Graph to the right:
1. 82 yxand
4053 yx
Graph:
2. )10(522 xy
and
)6(317 xy
AlgebraGraphing Inequalities 7.6Name________________________ Period _____
Write a system of inequalities for each problem below.Solve and graph each pair of inequalities.
3. Brian needs to buy two types of toys for hiscousins’ Christmas presents. Toy cars cost$4 and toy action figures cost $8. He wantsto buy at least four toys, and he can spendup to $40. Graph the solution and list allthe possible combinations of toys he canbuy.
Equations: _______________
_______________
4. Michelle works at two jobs. She makes $4an hour babysitting, and $6 an hour work-ing at the grocery store. She wants tomake more than $48 a week, but she has towork less than 11 hours a week.
Equations: _______________
_______________
List 3 ways she can work <11 hours and make > $48:
_________________________________________________________________________
AlgebraTest Review 7.7Solve each system of equations below using Substitution or Elimination.
100. 2223 yx 200. yx 3 62 yx 923 xy
300. 323 yx 400. 1552 xy 1532 yx 932 yx
Practice:Write a system of equations and solve:
500. Kenny sold pens and pencils at his school store. Pencils cost $.25, penscost $.35. In one morning he made $5.80 selling 20 pens and pencils.How many of each did he sell?
600. A juice company is combining fruit juices. The cranberry juice they areadding is 65% juice. They are mixing 45% apple juice to make 120 gal-lons of juice. How much of each should be mixed to create a mixture thatis 60% juice?
Practice:300. Use a graphing calculator to solve the system of equations below:
(round solution to the thousandth)
14.3215.2 xy and 02.306. xy
Practice:1000. Solve using a system of inequalities and graphing.
A small pizza costs $8 and large pizza costs $10. The small pizza uses 4ounces of dough and the large pizza uses 6 ounces. You have 60 ouncesof dough, and you want to sell at least $110 worth of pizzas. What is thegreatest number of large pizzas you can make and still make at least$110?
AlgebraPractice Quiz: Systems 7.7Name________________________ Period _____
Graph the system of linear equations below to find a solution:
1. 2423 xyxy 2
2. 13 yx12 xy
Solution 1. _________ Solution 2. __________Graph the system of inequalities below:
3. 22 xy1 xy
Solve each system of equations below using substitution, elimination, or a graphing calculator.(round to the hundredth where applicable)
4. 33 xy7 xy
4. ____________
5. 3.923.3 xy302.2045. xy
5. ____________
AlgebraPractice Quiz: Systems 7.7Name________________________ Period _____
Solve each system of equations below:
5. 334 yx423 yx
5.____________
6. xy 4210 xy 33
6.____________
Solve each:
7. Kayla spent one hour ironing shirts and pants. It takes her 5 minutes to iron a shirt
and only 3 to iron a pair of pants. If there were 16 items in the laundry, how many
were shirts and how many were pants?
7. s=_______ p=_______
8. How much of a 15% saltwater solution should be added to a 25% saltwater solution to
make a 10-liter solution of 22% saltwater?
8. 15%_______ 25%______
9. A company is mixing a blend of two different coffees. The first kind (x) costs $8 a
pound, and the second (y) costs $5 per pound. How much of each should they use
if they want 60 pounds worth $6.25 per pound?
9. x=_______ y=_______
10. Michelle scored 30 points by making 13 shots from the floor in a basketball game.
How many 2 and 3 pointers did she make?
10. 2s=_______ 3s=_______
11. In a cage full of bugs, there are beetles (6 legs) and spiders (8 legs). You count 30 bugs
and 192 legs. How many spiders and beetles are there?
11. Spiders_______ Beetles______
AlgebraPractice Quiz: Systems (4)Name________________________ Period _____
Solve each system of equations below:
5. 334 yx423 yx
5.____________
6. xy 4210 xy 33
6.____________
Solve each:
7. Kayla spent one hour ironing shirts and pants. It takes her 5 minutes to iron a shirt
and only 3 to iron a pair of pants. If there were 16 items in the laundry, how many
were shirts and how many were pants?
7. s=_______ p=_______
8. In a cage full of bugs, there are beetles (6 legs) and spiders (8 legs). You count 30 bugs
and 192 legs. How many spiders and beetles are there?
8. Spiders_______ Beetles______
9. A company is mixing a blend of two different coffees. The first kind (x) costs $8 a
pound, and the second (y) costs $5 per pound. How many pounds of each should
they use if they want 60 pounds of coffee that costs $375?
9. x=_______ y=_______
10. Michelle scored 30 points by making 13 shots from the floor in a basketball game.
How many 2 and 3 pointers did she make?
10. 2s=_______ 3s=_______
AlgebraDivisibility Rules 9.0Name________________________ Period _____
A Prime Number is a whole number whose only factors are 1 and itself. To find all of theprime numbers between 1 and 100, complete the following exercise:
1. Cross out 1 by Shading in the box completely.1 is neither prime nor composite. It has only 1 factor - itself.
2. Use a forward Slash \ to cross out all multiples of 2, starting with 4.2 is the first prime number.
3. Use a backward Slash / to cross out all multiples of 3 starting with 6.4. Multiples of 4 have been crossed out already when we did #2.5. Draw a Square on all multiples of 5 starting with 10. 5 is prime.6. Multiples of 6 should be X’d already from #2 and #3.7. Circle all multiples of 7 starting with 14. 7 is prime.8. Multiples of 8 were crossed out already when we did #2.9. Multiples of 9 were crossed out already when we did #3.10. Multiples of 10 were crossed out when we did #2 and #5.
All of the remaining numbers are prime.
How many prime numbers are left between 1 and 100? _____
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Answer: use your chart for help.
Is 51 prime? If not, what are its factors? ____________
Is 59 prime? If not, what are its factors? ____________
Is 87 prime? If not, what are its factors? ____________
Is 91 prime? If not, what are its factors? ____________
AlgebraDivisibility Rules 9.0Name________________________ Period _____
There are some easy tricks you can use to determine if a number is divisible by 2, 3, 4, 5, 6,8, 9 and 10.
A number is divisible by:2 - if it is even.3 - if the sum of its digits is divisible by 3.4 - if the number formed by the last 2 digits is divisible by 4.5 - if the ones digit is 5 or 0.6 - if it is divisible by 2 AND 3. (All even multiples of 3.)7 - there is no good trick for 7.8 - if the number formed by the last 3 digits is divisible by 8.9 - if the sum of the digits is divisible by 9.10 - if the last digit is a 0.11: We will learn this trick separately.
Practice: Write yes or no in each blank.
Determine whether 21,408 is divisible by:2 - ____ 6 - ____3 - ____ 8 - ____4 - ____ 9 - ____5 - ____ 10 - ____
Determine whether 1,345,866 is divisible by:2 - ____ 6 - ____3 - ____ 8 - ____4 - ____ 9 - ____5 - ____ 10 - ____
Determine whether 222,222,225 is divisible by:2 - ____ 6 - ____3 - ____ 8 - ____4 - ____ 9 - ____5 - ____ 10 - ____
Write the complete prime factorization for each number below. Use a factor tree if necessary:Ex: 1,600 1. 210 2. 297 3. 192
26 52
AlgebraGCF and LCM 9.0The GCF is the Greatest Common Factor between two or more numbers.
Sometimes the GCF is obvious:Find the GCF for each pair of numbers.1. 50 and 75 2. 49 and 56 3. 45 and 60
When the GCF is not obvious:Ex.Find the GCF between 405 and 585.
53333405
13533585 Common factors are 45533 , the GCF is 45.
notes:The GCF between a pair or set of numbers is the productof their common prime factors.
Practice:Find the GCF.
1. 108 and 126 2. 154 and 210 3. 108 and 288
The LCM is the Least Common Multiple. This means the smallest numberthat both numbers divide with no remainder.
The LCM is rarely obvious:Find the LCM for each pair of numbers.1. 5 and 7 2. 10 and 15 3. 16 and 24
When the LCM is not obvious:Ex.Find the LCM between 144 and 168.
332222144 Shared factors are ...3222 ,
73222168 other factors are ...732 so
GCF = 008,17323222
AlgebraGCF and LCM: Venn Diagrams 9.0
Venn Diagrams are a great way to solve GCF and LCM problems.
Example: Use a Venn diagram to find the GCF and LCM between 84 and 140.
732284 7522140
722 3 5
84 140
Review Practice:Find the GCF and LCM for each:
1. 36 and 168 2. 28, 42, and 105
Practice: Use a Venn diagram to find the GCF and LCM for each.
1. 45 and 60 2. 80 and 112 3. 28, 42, and 105
Example: Use a Venn diagram to find the GCF and LCM for 75 and 105:
AlgebraGCF and LCM 9.0Name________________________ Period _____
Find the GCF and LCM for each pair or set of numbers:You may use a calculator, and Venn diagrams are encouraged but not required.
1. 54 and 80 2. 88 and 136
GCF _____ LCM _______ GCF _____ LCM _______
3. 90 and 105 4. 45 and 72
GCF _____ LCM _______ GCF _____ LCM _______
5. 96 and 160 6. 153 and 180
GCF _____ LCM _______ GCF _____ LCM _______
7. 20 and 40 8. 64 and 88
GCF _____ LCM _______ GCF _____ LCM _______
9. 270 and 351 10. 143 and 221 (neither one is prime)
GCF _____ LCM _______ GCF _____ LCM _______
AlgebraFactoring the GCF 9.1You can find the GCF of expressions which include variables and exponents:
Examples:
1. Find the GCF of 2572 yx and
73120 yx :
2. Find the GCF of 11393 ba and
75124 ba :
Practice: Find the GCF for each pair or set:
1. 1015x and 2.
4378 nm and 3. ba 228 and
2025x nm 5130
221ab and
2230 ba
We have learned to Factor. Factoring is like Reverse Distribution.
To factor an expression:a. Look for the GCF of all terms, including the variables.b. Place the GCF outside of the parenthesis.c. Divide each original term by the GCF to get the terms inside the
parenthesis.
Examples: Factor each.
1. xyx 15257 2 2. 515105 132110 yxyx
Practice: Factor each.
1. 223 96160 yxyx 2.
234 154525 xxx
Practice: Factor each.
1. xyyxyx 139165 23 2. 210740 14444 yxyx
AlgebraFactoring the GCFName________________________ Period _____
For each polynomial, factor the GCF from the expression.These should be easy enough to factor the GCF in your head.
1. xx 2418 3 1. __________________________
2. 223 963 abbaa 2. __________________________
3. baba 203216 3. __________________________
4. 235 122814 mmm 4. __________________________
5. babaa 2223 14810 5. __________________________
6. 222 12188 xyxyx 6. __________________________
7. 234 152030 xyxyxy
7. __________________________
AlgebraFactoring the GCFName________________________ Period _____
For each polynomial, factor the GCF from the expression.You will likely need to find the GCF separately with these problems.
8. xx 6480 3 8. __________________________
9. 223 606852 abbaa 9. __________________________
10. 22 425670 baba 10. __________________________
11. 235 1084860 mmm 11. __________________________
12. 12689 7095 baba 12. __________________________
13. 222 765738 xyxyx 13. __________________________
AlgebraPolynomials 9.1A polynomial is a sum of one or more terms called monomials.
Examples: yx yxyx 42 3 312 x
A monomial is the product of variables and constants (numbers).
Examples: 32abc
7x x75
A binomial is the sum of two monomials.
Examples: aa 22 32 x 24 5
32 xx
A trinomial is the sum of three monomials.
Examples: cba 2 357 yx
The degree of a monomial is the sum of the exponents of its variables.
Examples: 22a 2nd degree cab65 = 1+6+1 = 8th degree
note: The degree of a constant is zero.
The degree of a polynomial is the largest degree of its monomial terms.
Examples: aa 72 2 is a 2nd degree binomial
cba 325 25 is a 5th degree trinomial
AlgebraPolynomials 9.1Ordering Polynomials:The general rule for ordering a polynomial is to write the terms in descendingorder by powers of a given variable:
Example: Arrange by descending powers of x: 53 352 xxx
Example: Arrange by descending powers of x: yxxyxy 2
Practice:Order the following polynomials by descending powers of x.
1. 232 235 xyxxxy 2.
32232 543 yxyxx
3. 2332 axaaxax 4.
5327 xyxx
Answer:What degree is each of the polynomials above?
Other tiebreakers: Alphabetical order.
Ex: Arrange by descending powers of a: cababaca 323323 352
Practice:Order the following polynomials by descending powers of a.
1. 222 23 bcaab 2.
233232 53 cababaa
3. yayaaxax 2332 4. aayaxaz 22 237
AlgebraMultiplying Polynomials 9.3Find the area of each rectangle below:
Multiplying Binomials:Setup a grid like the one above to solve the following:
1. )5)(3( xx 2. )2)(52( xx
The FOIL Method:First acOuter adInner bcLast bd
Examples: Expand each using the FOIL Method.
1. )3)(1( aa 2. )3)(2( yxyx
Practice: Expand each using the FOIL Method.
1. )5)(3( xx 2. )3)(2( caca
3. )3)(21( aa 4. )4)(( 22 xyx
35
4
3x
xyx
bdbcadacdcba ))((
AlgebraMultiplying Polynomials 9.3Find the area of each rectangle below:
Multiplying longer polynomials is easier using the grid method:Setup a grid like the one above to solve the following:Remember to combine like terms and place answers in descending order.
1. )5)(3( yyx 2. )2)(352( baba
Practice:Expand each.
1. )5)(3( yyx 2. )2)(352( baba
Practice:Express the area of the shaded region below as a polynomial in simplest form:
y
)3)(2( yxyx
432 ba
)432)(34( baba
x 3
9x
5x
AlgebraBeyond the GridOnce you have learned the grid and FOIL methods, you should beginto see multiplying polynomials is just distribution.
Examples:
1. )52)(( yxyx 2. )132)(45( baba
Practice:Multiply each.
1. )5)(2( yxyx 2. )432)(2( baba
Now, multiply each using the Distributive Property.You should notice something about the answers.
1. )5)(23( yxyx 2. )2(6)2( babbaa
3. )5(2)5(3 yxyyxx 4. )2)(6( baba
Work Backwards: write each as a product of binomials.
1. )32()32( xyxx 2. )32(2)32( babbaa
3. )3(5)3(2 yxyxx 4. )7(5)7(2 abaa
Use the grid method when problems get more complex:
1. )232)(52( yxyx 2. )23)(52( 223 aaaaa
AlgebraSpecial Cases 9.4Multiply each pair of binomials using the FOIL Method.Simplify answers.
1. )5)(3( xx 2. )32)(5( baba
3. )3)(3( xx 4. )5)(5( baba
5. )3)(3( xx 6. 2)5( ba
#1 and #2 are typical trinomials.#3 and #4 are called DIFFERENCE OF SQUARES. Why?#5 and #6 are called PERFECT SQUARE trinomials.
More practice: Difference of Squares.Solve each using FOIL, try to recognize a shortcut.
1. )32)(32( xx 2. )32)(32( baba
3. )5)(5( 22 xxxx 4. )3)(3( 33 aaHow do you recognize a difference of squares?
More practice: Perfect Squares.Solve each using FOIL, try to recognize a shortcut.
1. 2)32( x 2.
2)32( ba
3. 22 )5( xx 4.
23 )3( a
Challenge: Expand ))()()(( babababa in 1 minute.
AlgebraQuiz Review 9.4Factor out the GCF for each trinomial.
100. 32223 102015 xyyxyx
200. 22424 36135108 bababa
300. axaxxa 4872136 23
400. xxx 18768119 23 Multiply each:Order your answers by descending powers of x or a.
100. )(3)3(2 2 xyxxyxxy
200. ))(2( 22 abba
300. )4)(35( 22 xxxx
400. )22)(3( cbacba Multiply each.Order your answers by descending powers of x or a.
100. )330)(330( 200. 2)5( ba
300. )3)(3( 3434 xxxx 400. 2222 )3()3( aa
AlgebraFactoring and FOIL Practice Quiz 9.4Name________________________ Period _____
Factor each expression (Reverse distribution):
1. 54233 302142 xyyxyx
1. _____________________
2. 2435 8048 baba 2. _____________________
3. 4526 1529557 xyxyx
3. _____________________
Simplify each(Distribute, combine like terms, and then reorder the terms by descending powers of x or a):
4. )3()(4 2 yxyxxyyx 4. _____________________
5. )54(2)3( abbaaab 5. _____________________
6. )7(2)2( 22 xxxyx6. _____________________
Multiply(FOIL or Grid method)
7. )3)(4( 2 xx7. _____________________
8. )2)(3( aa 8. _____________________
9. ))(23( yxx 9. _____________________
AlgebraFactoring and FOIL Practice Quiz 9.4Name________________________ Period _____
Multiply each (Look for perfect squares and difference of squares):
10. ))(( yxyx 10. _____________________
11. 2)32( a
11. _____________________
12. )23)(23( aa 12. _____________________
13. )2)(2( 33 yxyx 13. _____________________
14. )2)(2)(2( bababa 14. _____________________
15. 2)23( x
15. _____________________
16. 2)1)(1( xx16. _____________________
AlgebraPolynomial ApplicationsA common use for multiplying polynomials involves finding area.
Example: Express the area of the shaded regions in terms of x.
9.4
I will call these ‘frame’ problems because the diagrams usually looklike frames.
Practice: Express the area of the shaded regions in terms of x.
x + 4
x +
1
x -3
2x +
5
2x + 5
x
x
x
x +
5
x + 8
x + 1
2x
33
x
x
3 3
AlgebraPolynomial ApplicationsWord problems can involve similar area problems, but the diagramsmust be given.
Example:You are matting a photograph that is twice as tall as it is wide. You want tohave five inches of matting around the entire photograph. Express the area ofmatting you will need based on the width (w) of the photograph.
Example:Barry bought a new rectangular rug for his rectangular dining room. The rugis three feet longer than it is wide. The room is six feet wider than his rug, andseven feet longer than the rug. Express the area of bare floor that will beshowing in terms of the rug’s width (w).
Answer: If there are 190 square feet of bare floor showing, what ar the di-mensions of the rug?
Practice:Jeremy has a backyard pool surrounded by a tiled walkway that is two yardswide. The pool is 5 yards longer than it is wide. Express the area of the walk-way in terms of the width (w) of the pool.
Answer: If the walkway is 196 square yards, how long is the pool?
Practice:A painting has a frame that is 7 inches wider and 8 inches taller than the art-work it surrounds. The artwork is 5 inches taller than it is wide. Express thearea of the frame in terms of the painting’s width (w).
Answer: If the area of the frame is 196 square inches, what is the height ofthe painting?
9.4
AlgebraPolynomial Applications 9.4Name________________________ Period _____
Express the area of each shaded region in terms of x.
1. 2. 3.62 x
x2
12 x
x13x11x
Express the area of each shaded region in terms of x.4. 5. 6.
3 7
x + 6
x +
10
2x -
1
x +
2
x x x
xx
37
6x
+ 2
55
x10
2x3 3
AlgebraPolynomial Applications 9.4Name________________________ Period _____
Solve each. Include a sketch for each.
7. Connor is planting a garden surrounded by 1-foot square concrete blocks. The garden will be 10feet longer than it is wide. Express the number of square blocks he will need based on the width (w)of the garden.
If he uses 56 blocks, how many square feet is the area enclosed by the blocks? ______
8. Kerry takes a sheet of paper that is 3 inches shorter than it is wide. He cuts a hole out of thepaper that leaves 2 inches of paper on all sides of the hole. Express the area of the remaining paperrectangle in terms of w, the width of the original sheet.
If there are 52in2 of paper remaining, what were the dimensions of the cut-out hole? ______
9. A company manufactures windows that are 30 inches taller than they are wide. The windowcomes with an aluminum frame that is 6 inches wide on three sides, and 10 inches wide at the bottom.Express the area of the aluminum frame in terms of the window’s width (w).
If the area of the frame is 1,392in2, what is the height of the window? ______
AlgebraStandard Form and Factoring 9.5A Quadratic Equation written as a function looks like this:
CBxAxy 2 We will call this Standard Form.
Examples: List values for A, B, and C:
532 2 xxy xxy 52
When you multiply a pair of (1st degree) binomials,you get a quadratic expression.
152)5)(3( 2 xxxxThink!In the equation above, what are the A, B, and C values?How did we get the values for B and C?
Factoring: Easy ones.Today we will learn to factor simple quadratics by reversing the FOIL method.
Review: Multiply )4)(2( xx
862 xx Factoring:Find two numbers which can be added to get 6and multiplied to get 8.
More Examples: Factor.
1. 652 xx 2. 1092 xx
AlgebraStandard Form and Factoring 9.5Practice: Factor. Write Prime for any that cannot be factored.
1. 1282 xx 2. 1582 xx
3. 24102 xx 4. 9102 xx
5. 3382 xx 6. 542 xx
7. 36162 xx 8. 452 xx
Practice: Factor. Write Prime for any that cannot be factored.
1. 782 xx 2. 3072 xx
3. 302 xx 4. 42192 xx
5. 122 xx 6. 432 xx
7. 1072 xx 8. 22 xx
AlgebraFactoring: GCF with ‘Easy Ones’ 9.5Examples: Factor completely.
Begin by factoring out the GCF.Finish by using reverse FOIL.
xxx 8102 23 yxyyx 24142 2
Practice: Factor Completely.
1. 60405 2 xx 2. 345 158 xxx
3. aaxax 24102 4. 90393 2 xx
5. xxx 72102 23 6. 2222 3013 yxyyx
Practice: Factor. Write Prime for any that cannot be factored.
1. 21243 2 xx 2. xxx 23 2
5. 142814 2 xx 6. 150655 2 xx
7. 90639 2 xx 8. yxyyx 482424 2
AlgebraFactoring ‘Easy Ones’ with GCFs 9.5Name________________________ Period _____
Factor eact expression by first factoring the GCF and then using reverse FOIL.Write Prime for any that cannot be factored.
1. 2444 2 xx 2. 98282 2 xx
3. 60355 2 xx 4. yxyyx 17182
5. 108369 2 xx 6. yxyyx 48126 2
7. xxx 1003010 23 8. 7042 2 xx
9. 42497 2 xx 10. 2062 2 xx
11. 2222 72yxyyx 12. 36279 2 xx
13. 363 2 xx 14. aaxax 31322
AlgebraFactoring: Difference of Squares 9.7Examples: Factor completely.
92 x 4925 2 x 364 xPractice: Factor Completely.
1. 12 x 2. 1219 2 x 3. 258 x
4. 169100 2 x 5. 19 4 x 6. 502 2 xYou can factor out the GCF first.
Examples: Factor completely.
xx 253 10016 2 x 273 4 xPractice: Factor Completely.
1. 455 2 x 2. xxy 364 2
3. xx 74. xyyx 433
5. xx 66 5 6. 644 2 x
AlgebraFactoring: Perfect Squares 9.7Examples: Factor completely.
962 xx 497025 2 xxPractice: Factor Completely.
1. 25102 xx 2. 64162 xx
3. 9124 2 xx 4. 4914 24 xx
Recognizing Perfect Squares:Don’t be fooled by these imposters!Only one is a perfect square. Can you find it? Factor and Check!
1. 49142 xx 2. 81184 2 xx
3. 42025 2 xx 4. 169 4 xx
Easy Ones, Perfect Squares, and Difference of Squares:Put it all together. Try to recognize how to factor each.
1. 40162 2 xx 2. 812 a
3. 562 xx 4. 513417 2 xx
AlgebraFactoring and FOIL ReviewName________________________ Period _____
Multiply(FOIL or Grid method)
1. )5)(13( 2 xxx1. _____________________
2. 2)34( a
2. _____________________
3. ))(23( yxx 3. _____________________
4. )712)(712( 33 xx4. _____________________
Factor Each Completely(Look for GCFs, Perfect Squares, and Difference of Squares.Write PRIME for any that cannot be factored.)
5. 30132 xx5. _____________________
6. 164 x6. _____________________
7. 2092 xx7. _____________________
8. xxx 183 23 8. _____________________
9. 42172 xx9. _____________________
10. 273 2 x10. _____________________
11. 25309 2 xx11. _____________________
12. 108393 24 xx12. _____________________
AlgebraFactoring PracticeName________________________ Period _____
Challenge 1:Factor Completely.
98256 yyx
Ch. 1. _______________________________________________
Challenge 2:The number 65,535 is equal to 216 - 1. Use what you know about a difference of squares to find thefour prime factors of 65,535 without a calculator (be ready to explain how this can be done).
Ch. 2. _______________________
AlgebraFactoring and FOIL Practice QuizName________________________ Period _____
Multiply(FOIL or Grid method)
1. )32)(( 2 baba1. _____________________
2. 2)52( x
2. _____________________
3. )3)(3( yxyx 3. _____________________
4. )32)(2( 3 xx4. _____________________
Factor Each Completely(Look for GCFs, Perfect Squares, and Difference of Squares.Write PRIME for any that cannot be factored.)
5. 30132 xx5. _____________________
6. 22121 ba 6. _____________________
7. 48192 xx7. _____________________
AlgebraFactor Each Completely(Look for GCFs, Perfect Squares, and Difference of Squares.Write PRIME for any that cannot be factored.)
8. yxyyx 1582 8. _____________________
9. 11122 xx9. _____________________
10. xx 155 2 10. _____________________
11. 164025 2 xx11. _____________________
12. 4022 2 xx12. _____________________
13. 814 x13. _____________________
14. 124144 2 xx14. _____________________
Factoring and FOIL Practice QuizName________________________ Period _____
Algebra
Algebra
Factoring and FOIL Self-Check 9.7Name________________________ Period _____
Factor each (Look for perfect squares and difference of squares, GCF, and easy ones).
1. 962 xx 2. yxyyx 652
3. 499 2 x 4. 3072 xx
5. 49284 2 xx 6. 644 2 x
Factoring and FOIL Self-Check 9.7Name________________________ Period _____
Factor each (Look for perfect squares and difference of squares, GCF, and easy ones).
1. 962 xx 2. yxyyx 652
3. 499 2 x 4. 3072 xx
5. 49284 2 xx 6. 644 2 x
AlgebraFactoring Review. 9.7Easy Ones: Factor completely. Write PRIME for any that cannot be factored.
Ex.: 2092 xx
1. 1662 xx 2. 2832 xx
3. 54252 xx 4. yxyyx 15123 2
Difference of Squares: Factor completely. Write PRIME where applicable.
Ex.: 916 2 x
1. 14449 2 x 2. 1002 x
3. 22 ayax 4. 364 2 x
Perfect Squares: Factor completely. Write PRIME where applicable.
Ex.: 25102 xx
1. 1682 xx 2. 100404 2 xx
3. 122 xx 4. 22 366025 yxyx
AlgebraHard Ones ‘Magic Number’ 9.7Look at the trinomial below.Is there a GCF to be factored?Is it an ‘Easy One’, a Perfect Square, or a Difference of Squares?
15164 2 xxThe answer to all of these questions is “No.” We will call this type of factoringthe ‘Magic Number’ Method.
Example: Factor 15164 2 xx1. Find/Factor the Magic Number.
2. Rewrite the middle term.
3. Regroup.
4. Factor out the GCFs.
5. Finish (___)(___)
Two more examples. Watch Carefully!
1. 20113 2 xx 2. 2910 2 xx
Practice: Factor each completely.
1. 239 2 xx 2. 10134 2 xx
Practice: Factor each completely.
1. 42025 2 xx 2. 27303 2 xx
AlgebraHard Ones ‘Magic Number’ 9.7Look at each trinomial below. DO NOT TRY TO FACTOR THEM.Label each with: EASY ONE
DIFFERENCE OF SQUARESPERFECT SQUAREHARD ONE (MAGIC NUMBER)
(hint: there are two of each)
1. 542 xx 2. 167281 2 xx
3. 1442 x 4. 9465 2 xx
5. 40222 xx 6. 922 yx
7. 495616 2 xx 8. 18253 2 xx
Now, try to factor them.
1. 542 xx 2. 167281 2 xx
3. 1442 x 4. 9465 2 xx
5. 40222 xx 6. 922 yx
7. 495616 2 xx 8. 18253 2 xx
AlgebraFactoring Review 9.7Factor each: Write Prime for any that cannot be factored.
1. 49142 xx 2. 2832 xx
3. 61110 2 xx 4. 2225 yx
5. 4129 2 xx 6. 29 x
7. 342 xx 8. 21228 2 xx
Name________________________ Period _____
AlgebraFactoring Review 9.7Factor each: Write Prime for any that cannot be factored.
9. 54212 xx 10. 77182 xx
11. 49144 2 xx 12. 22 12181 yx
13. 51112 2 xx 14. 814 22 yx
15. 22 252 xxyy 16.
4224 2 yyxx
Name________________________ Period _____
AlgebraFactoring Quiz Review 9.7Factor each: Write Prime for any that cannot be factored.
100. 162 x 200. 5432 xx
300. 122512 2 xx 400. 22 9216 yxyx
Factor each: Write Prime for any that cannot be factored.
100. 2169 x 200. 48192 xx
300. 26 xx 400. 992 22 xyyx
Factor each: Write Prime for any that cannot be factored.
100. 497 x 200. 11025 2 xx
300. 81110 2 xx 400. 45 24 xx
AlgebraFactoring and FOIL Practice Quiz 9.8Name________________________ Period _____
Multiply each (Look for perfect squares and difference of squares,order the terms by descending powers of x):
1. 2)3( x
1. _____________________
2. )13)(13( aa2. _____________________
3. )2)(3( xyyx 3. _____________________
Factor each COMPLETELYNONE OF THE PROBLEMS BELOW ARE PRIME.(Look for perfect squares and difference of squares, easy ones and hard ones).
4. 229 yx
4. _____________________
5. aa 93 5. _____________________
6. 60172 xx6. _____________________
7. 12102 2 xx7. _____________________
AlgebraFactoring and FOIL Practice Quiz 9.8Name________________________ Period _____
Factor each COMPLETELYWrite PRIME for any that cannot be factored.(Look for perfect squares and difference of squares, easy ones and hard ones).
8. 22 10025 yx
8. _____________________________
9. 81364 2 xx9. _____________________________
10. 4032 xx10. _____________________________
11. 143910 2 xx11. _____________________________
12. 4224 2 yyxx
12. _____________________________
AlgebraSimplifying ExpressionsPractice: Factor each.
1. xx 32 2. 92 x 3. 152 2 xxNow, try to simplify the following:
1. 93
2
2
xxx
2. 1529
2
2
xx
x3. xx
xx3
1522
2
Practice: Simplify each expression.
1. )2)(7()7(
xx
xx2. 20
2092
2
xxxx
Practice: Simplify each expression.
1. 251025
2
2
xx
x2. 492
452
2
xxxx
Practice: Simplify each expression.
1. )3)(3()3)(3(2
xxxxxx
2. 263105
2
23
xxxx
3. 2520420236
2
2
xxxx
4. 968118
2
24
xxxx
AlgebraSimplify by Factoring 9.7Factor each and simplify where possible.
1. 2)7()7(
xxx
2. 665
2
2
xxxx
3. 121312
2
2
xxxx
4. 251025
2
2
xxx
5. 1215312153
2
2
xxxx
6. 7835405
2
2
xxxx
7. xxxxxx
44149
23
23
8. 2345
2
24
xxxx
Name________________________ Period _____
AlgebraSolving Equations by Factoring 9.8Practice: Solve each.
1. 553 x 2. 15)5(3 x 3. 0)3( xx
If 0ab then either 0a or 0b .
If 0)5)(3( xx then either 0)3( x or 0)5( x .
Examples: Solve each for x. Each will have two solutions.
1. 0)7( xx 2. 0)5)(9( xx
3. 01662 xx 4. 01572 2 xx
Practice: Solve for x. Each will have two solutions.
1. 032 xx 2. 02092 xx
3. 0162 x 4. 01076 2 xx
Tricky Examples: Solve each for x.
1. 22 xx 2. xx 25
51 2
Tricky Practice: Solve each for x.
3. 1032 xx 4. 12
21 2 xx
AlgebraSolving Quadratics by Factoring 9.7Factor each and simplify where possible.
1. 0)5)(3( xx 2. 0252 2 xx
3. 036122 xx 4. 0125 2 x
5. 0122 xx 6. 010196 2 xx
7. 0239 2 xx 8. 762 xx
9. xx 618 2 Challenge: 24 109 xx (4 solutions)
Name________________________ Period _____
AlgebraFactoring ProblemsFor the problems below, you must know the Pythagorean Theorem:
In any right triangle:
a2 + b2 = c2
Example:Find the lengths of the sides of the right triangle below.
Practice:Find the lengths of the sides of the right triangle below.
a
b
c
x
2x+2
2x+3
x
3x+3
3x+4
Practice:
1. In a right triangle, the hypotenuse is 9 inches longer than the shortest side.The length of the medium side is just one inch longer than the length ofthe shortest side. What is the perimeter in inches of the triangle?
2. The hypotenuse of a right triangle is 1cm longer than the long leg. Theshort leg is 1cm shorter than half the long leg. What is the triangle’sarea?
AlgebraSolving Quadratics by FactoringFactor each and simplify where possible.
1. 021102 xx 2. 0916 2 x
3. 562 xx 4. xx 20325 2
5. 1099 2 xx 6. 1032 xx
7. 5188 2 xx 8. 043 xx (3 solutions)
9. The equation 0362 kxx has only one solution for positive integer k. What is k ?
10. Find the perimeter of the triangle below.
Name________________________ Period _____
x
5x+5
5x+6
AlgebraClever Factoring:Some tricks and more difficult problems:
Example:
One of the solutions to the equation xax 62 is 5.a. What is the value of a?b. What is the other solution?
Practice:
1. The equation 83 2 axx has 4x as a solution.a. What is the value of a?b. What is the other solution?
2. The equation 252 xax has 1x as a solution.a. What is the value of a?b. What is the other solution?
Example:
How can the polynomial 9)( 2 yx can be factored into theproduct of two trinomials?
Practice:
1. Factor the following into a product of trinomials: 22)2( yx .
2. Factor the following into a product of trinomials: 251022 xyx .
Solving Trickier Equations Practice:
1. Solve for x: 0)4()4(3 22 xxx .
Hint: Where have you seen something similar to this before?
2. Solve for x: 029102
xx
.
Hint: use a common denominator.
3. Solve for x: 2
33146
xxx
.
AlgebraFactoring Test Review 9.8Perfect Squares and Difference of Squares: Factor each.
100. 29 x 200. 246 2 x
300. 18122 22 xyyx 400. 814 x
Easy Ones and Magic Number:Write Prime for any that cannot be factored.
100. 862 xx 200. 72222 xx
300. 3512 2 xx 400. 22 26 yxyx
Solve each: Write Prime for any that cannot be factored.
100. 030
31 2 xx
200. xx 11242
300. xx 47
74 2
400. xxx 14256 23
AlgebraFactoring and FOIL Practice Test 9.8Name________________________ Period _____
Factor each COMPLETELYWrite PRIME for any that cannot be factored.(Look for perfect squares and difference of squares, easy ones and hard ones, and GCF problems).
1. 22 4 yx
1. _____________________
2. xx 8534 2 2. _____________________
3. 30132 xx3. _____________________
4. 103 2 xx4. _____________________
5. aaa 44 23 5. _____________________
6. 24192 2 xx6. _____________________
7. 8011 24 xx7. _____________________
AlgebraFactoring & FOIL Practice Test (4) 9.8Name________________________ Period _____
Solve for x:Some problems may have more than one solution. List all solutions in the blank provided.
8. 0)7(3 xx8. x=_____________________
9. 0962 xx9. x=_____________________
10. xx 17610 2 10. x=_____________________
11. xx 18722 11. x=_____________________
Multiply:
12. )5)(45( xx12. _____________________
13. 22 )53( xx
13. _____________________
14. )2)(3)(2( xxx14. _____________________
AlgebraFactoring and FOIL Practice Test 9.8Name________________________ Period _____
Solve for x:Some problems may have more than one solution. List all solutions in the blank provided.
8. 0)7(3 xx8. x=_____________________
9. 0962 xx9. x=_____________________
10. xx 17610 2 10. x=_____________________
11. xx 31261 2
11. x=_____________________
Simplify each:
12. 665
2
2
xxxx
12. _____________________
13. 81481364
2
2
xxx
13. _____________________
14. 205104
2
23
xxx
14. _____________________
EOC REVIEW26 Questions EOC Review #1Solve each: Give the BEST Answer. You may use a graphing calculator.
1. Which quadrant contains the vertex of the following: 1182)( 2 xxxf
a. 1st b. 2nd c. 3rd d. 4th_______
2. What type of equation is described by the data below?
x -2 0 2 4 6
f(x) -1 -5 -1 11 31
a. Linear b. Quadratic c. Exponential d. None of these._______
3. The equation )11(53 xy passes through which of the following points?
a. (-3, 11) b. (3,-11) c. (-11, -3) d. (-11, 3)_______
4. Solve the following equation for x: bxyax
a. ybax
b. ybax
c. ba
yx
d. ba
yx
_______
5. What is the 100th term in the following sequence: 25, 36, 49, 64...
a. 10,404 b. 10,609 c. 10,816 d. 11,025_______
6. To download music from the web, an internet site offers a monthly membership and charges$0.59 a song. If the monthly membership is $15, which equation represents the cost (c) of buyingx songs in one year with the club?
a. c=0.59x+15 b. c=15x+0.59 c. c=0.59x+180 d. c=0.59x-180
_______
7. Which parabola below would have the narrowest graph?
a. xxy 457 2 b. xxy 457.0 2 c. xxy 745 2 d. xxy 771 2
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #18. Multiply: )2)(53( baba
a. 22 4135 baba b. 22 575 baba c. 22 576 baba d. 22 576 baba
_______9. What is equation for a horizontal line which passes through (-2, -3)?
a. x=-2 b. x+3=0 c. y=-2 d. y+3=0
_______
10. Which equation below has a graph with a slope of -½?
a. x-2y=12 b. -2x+y=12 c. -y-2x=12 d. x+2y=12_______
11. How many solutions are there to the equation: 0273 2 x
a. 0 b. 1 c. 2 d. Infinite_______
12. What is the equation for a line passing through (-2, 5) perpendicular to 83 xy ?
a. )2(315 xy b. )5(32 xy c. )2(
315 xy d. )2(35 xy
_______
13. If the equation 72 yx were graphed, which of the four quadrants would be shaded completely?
a. 1st b. 3rd c. 4th d. None._______
14. What is the range for the function xxxf 4)( 2 ? hint: find the vertex.
a. }4{ y b. }4{ y c. }4{ y d. {all real numbers}
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #115. Factor Completely: xxx 422 23
a. )2(2 2 xxx b. )2)(22( xxx c. )2)(1(2 2 xxx d. )2)(1(2 xxx_______
16. Which formula could be used to find the nth term of the sequence below?
160, 80, 40, 20, 10, …
a. n
an
160 b.
nan 2
320 c. nna
2160
d. 12160
nna
_______
17. Write an equation based on the table below showing the amount earned a mowing lawns based on thenumber of hours worked h, including an initial fee.
hours worked h 1 5 7 9 11earned amt. a $14.75 $53.75 $73.25 $92.75 $112.25
a. 25.550.9 ha b. 575.9 ha c. 5.525.9 ha d. 75.59 ha_______
18. In problem number 17 above, how much could you earn mowing lawns for 6 hours and 15 minutes?
a. $64.96 b. $65.45 c. $65.94 d. $66.24_______
19. Which equation below does NOT represent a function?
a. y=x b. x=y2 c. y=1 d. y=x2
_______
20. The height of a flare fired from a gun can be described by: tth 6016 2 where t is the time inseconds and h is the height in feet. How long will it take for the flare to reach 36 feet?
a. .75 seconds b. 1 second c. 1.5 seconds d. 3 seconds
_______
21. Solve for x: 023 2 xx
a. }1,32{ xx b. }1,
32{ xx c. }1,
32{ xx d. no solutions
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #122. A photograph is two inches taller than it is wide. The frame around the photo is three inches wide.Which expression below represents the area of the frame based on the width of the photo?
a. 12w b. 6w+24 c. 12w+48 d. w2+12w
_______
23. Which equation below is parallel to 932 yx and shifted up 5 units?
a. 232
xy
b. 1432
xy
c. 1432
xy
d. 223
xy
_______
24. A basketball is dropped from a height of 120 feet. Each time it lands it bounces ¾ of the heightit reached the last time. How high does the ball reach after the 5th bounce?
a. 50.6 ft b. 38.0 ft c. 28.5 ft d. 21.4ft_______
25. If y varies directly as x, and when y=6, x=15, solve for y when x=20.
a. 50 b. 8 c. 7.5 d. 4.5
_______
26. Solve the following system of equations: 723 yx232 yx
a. (-1, 5) b. (5, -1) c. (-4, 19) d. (5, -4)
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #2Solve each: Give the BEST Answer. You may use a graphing calculator.
1. Divide the following: 3
5
105.1109.3
a. 8106.2 b. 2106.2 c. 2106.2 d. 26_______
2. The height of a baseball struck at 45 meters per second can be described by tth 451.9 2 . Howhigh will the ball be after 2 seconds?
a. 71.8 meters b. 53.6 meters c. 35.9 meters d. 126.4 meters_______
3. Write an equation based on the table below showing the cost c of a cab ride based on the number ofmiles driven m.
cost c 4.30 5.55 8.05 14.30 15.55
Miles m 1 2 4 9 10
a. mc 25.1 b. 30.13 mc c. mc 25.105.3 d. 25.105.3 mc
_______
4. In problem number 3 above, which value represents the dependent variable?
a. Miles driven b. Cost of cab ride c. y-intercept d. $1.25_______
5. A car drives up a mountain for 14 miles, and gains 3,700 feet in altitude. What is the approximateslope of the road? (1 mile=5,280 feet)
a. 201
b. 101
c. 200
1d. 10
_______
6. A particular species of shark weighs 12 pounds at birth, and gains 3 pounds per week until it is 3 yearsold. Which of the following equations could be used to find the weight y of a young shark who is x weeksold?
a. y=12x+3 b. y=3x c. y=3x+12 d. y=3x-12
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #27. Where does the graph of 52 xy cross the y-axis?
a. 5 b. 0 c. 3 d. 4
_______
8. A bakery can make 30 batches of chocolate chip cookies in 480 minutes, and 40 batches in 600 minutes.After the initial time required for preparation, how long does it take to bake each batch of cookies?
a. 16 minutes b. 15 minutes c. 12 minutes d. 10 minutes
_______
9. What is equation for a line with a slope of zero which passes through (-3, 2)?
a. x=2 b. x=-3 c. y=-3 d. y=2
_______
10. Which equation below has a graph with an undefined slope?
a. x=2y b. y=0 c. x-5=0 d. y-x=0_______
11. A player scored 37 points, making 16 shots from the field.How many of these shots were three-pointers?
a. 11 b. 0 c. 5 d. 8_______
12. Mark earns $20,000 per year, and an additional amount equal to 1%of his total sales. Which equationbelow could be used to graph Mark’s salary (y) based on his sales (x) ?
a. 000,2001. xy b. 000,20 xy c. 000,2001. xy d. xy 000,20
_______
13. Which equation below represents a line which passes through the points (-3, 3) and (3, 5)?
a. 631
xy b. 431
xy c. 431
xy d. 631
xy
_______
14. What is the range for the function 52)( 2 xxf for the domain }5{ xD
a. }55{ yR b. }45{ yR c. }55{ yR d. }45{ yR
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #215. Which of the following is a factor of: 7116 2 xx
a. 12 x b. 12 x c. 73 x d. 16 x_______
16. Which formula could be used to find the nth term of the sequence below?
7, 14, 28, 56, 112, …
a. nan 7 b. 27nan c. nna 7 d. )2(7 1 n
na_______
17. A unit cube has edges that are 1 unit long, so that the surface area of a unit cube is 6u2. Which formulabelow could be used to find the surface area A of a stack of unit cubes that is n cubes tall?
1 2 3 ...n
(all answers in units2)
a. 26nA b. 24 nA c. nnA 24 2 d. )2(4 nA
_______
18. For
8273
a and
15
04b find ba 2 .
a.
1511410
b.
1791410
c.
179142
d.
151142
_______
19. Which equation below does NOT represent a function?
a. xy 2 b. 2y c. yx 2 d. 2x
_______
20. The height of a flare fired from a gun can be described by: tth 6016 2 where t is the time inseconds and h is the height in feet. How long will it take for the flare to reach its peak height?
a. 211 seconds b.
851 second c.
871 seconds d.
433 seconds
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #221. Solve for x: 023 2 xx
a. }1,32{ xx b. }1,
32{ xx c. }1,
32{ xx d. no solutions
_______
22. A dining room is five feet longer than it is wide. You purchased a rug that fits in the room, leaving 2feet of bare floor around all four sides of the rug. Which expression below represents the area of the rugbased on the width of the room?
a. w(w-4) b. (w-4)(w+1) c. (w+4)(w-1) d. (w+5)(w+1)
_______
23. Which equation below is parallel to 1032 yx but is shifted three units to the right ?
a. 1632 yx
b. 2032 yx
c. 1632 yx
d. 2032 yx_______
24. What is the distance between the following points on the coordinate plane? (-2, 5) (6, -1)
a. 5 units b. 24 units c. 54 units d. 10 units_______
25. Find the midpoint of segment AB for A = (9,2) and B = (-1, -7)
a. (4, -4.5) b. (4, -2.5) c. (5, -4.5) d. (5, -2.5)
_______
26. The center of a circle drawn on the coordinate plane is at (4, -9).If one end of a diameter AB is at A(-3, 7), what are the coordinates of B?
a. (11, 23) b. (-10, -25) c. (11,-25) d. (-10, 23)
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #31. Simplify: 14
32
yxyx
a. 42 yx b. 22
1yx c. 2
4
xy
d. 2
2
xy
_______
2. The height of a rocket launched at 30 meters per second can be described by tth 301.9 2 .How high will the rocket be after 1.5 seconds?
a. 31.4 meters b. 53.6 meters c. 231.3 meters d. 24.5 meters
_______
3. Write an equation based on the table below showing the cost c of a stereo rental based on the days d itis rented:
cost c $22 $34 $58 $118 $130
days d 1 2 4 9 10
a. dc 22 b. 1210 dc c. 1012 dc d. 184 dc
_______
4. In problem number 3 above, which value represents the dependent variable?
a. Days Rented b. Slope c. Cost of Rental d. Cost Per Day_______
5. A ski slope drops 1,400 feet. From start to finish, the skier travels about 2 miles horizontally.Approximately what is the average slope of the mountain?(1 mile=5,280 feet)
a. 152
b. 71
c. 751
d. 501
_______
6. A youth group sells cookies for $8 a box. If they spent $1000 buying the cookies, which equation belowshows the profit (p) made by the group after selling (b) boxes?
a. p=8b-1000 b. p=8b+1000 c. p=1000b+8 d. p=8b
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #37. Where does the graph of 255 xy cross the x-axis?
a. 0x b. 1x c. 2x d. 3x
_______
8. Multiply: 2)2( yx
a. 22 2yx b. 22 4yx c. 22 4 yxyx d. 22 44 yxyx
_______
9. For
5974
a and
23
15b find ba 3 .
a.
110419
b.
10411
c.
11181019
d.
1181011
_______
10. Which equation below has a graph with a slope of ¾?
a. 3x - 4y=12 b. 3y - 4x =12 c. 3y = 4x + 12 d. both a and c_______
11. A store has a total of 20 three-wheelers and four wheelers (off-road vehicles). If they have a total of65 wheels on all the vehicles, how many three-wheelers do they have?
a. 5 b. 10 c. 15 d. 20_______
12. What is the equation for a line passing through (-3, -1) and (3, 3) ?
a. 323 xy b. 323 yx c. 323 yx d. 323 yx
_______
13. Which of the following is a solution to the system of inequalities below:
73 xy and 1223 xy
a. (10, -10) b. (-10, -10) c. (10, 10) d. (-10, 10)
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #314. What is the 29th term in the following sequence: 15, 11, 7, 3, ….
a. -89 b. -93 c. -97 d. -101
_______
15. Solve for x: 50152 xx
a. }10{ x b. }5{ x c. }5,10{ xx d. }5,10{ xx_______
16. The number of bacteria present in an experiment can be approximated by ttn 10006.5 2 ,where t is the time in minutes. How many bacteria will there be in one-half hour?
a. 501.4 b. 35,040 c. 30,168 d. 58,224
_______
17. Write an equation based on the table below showing the amount earned a babysitting based on thenumber of hours worked h, including an initial fee.
hours worked h 3 5 7 9 11earned amt. a $21 $31 $41 $51 $61
a. 5.45.5 ha b. 36 ha c. 56 ha d. 65 ha_______
18. In problem number 17 above, what does the y-intercept represent?
a. Hours worked b. Total Earnings c. Initial fee d. Hourly wages_______
19. The steepest section of the Tour De France climbs 300 meters over the course of 11 kilometers.What is the average slope of the course in this section? (1km = 1000m)
a. 113
b. 110
1c.
1103
d. 11
300
_______
20. What are the coordinates of the vertex of the following: xxy 183 2
a. (-3, 27) b. (3, 18) c. (3, -27) d. (-3, -18)
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #3Name________________________ Period _____
21. Solve for x: 1092 2 xx
a. }2{ x b. }4.0{ x c. }5.2,2{ xx d. }4.0,2{ xx_______
22. What is the range for the function 9)( 2 xxf for the domain }2{ x . (hint: pick several values)
a. }9{ y b. }5{ y c. }5{ y d. }95{ y
_______
23. What would happen to the equation 92 xy if it were changed to 721
xy ?
a. The line would shift up two units and the slope would be parallel to the original.
b. The line would shift down two units and the slope would be parallel to the original.
c. The line would shift up two units and the slope would be perpendicular to the original.
d. The line would shift down two units and the slope would be perpendicular to the original.
_______
24. About how many years will it take $400 invested at 6% annual compound interest to double in value?trpA )1(
a. 10 years b. 12 years c. 15 years d. 20 years_______
25. Solve the following system of equations: xy 2 1223 xy
a. (-3, -6) b. (6, -3) c. (3, 6) d. (-3, 6)
_______
26. Which formula below could be used to determine the nth term in the following sequence:
1, 3, 6, 10, 15, …
a. 1)1( nnan b. 2
)1(
nnan c. 2
)1(
nnan d. )1(2 nnan
_______
EOC REVIEW26 Questions EOC Review #41. Simplify: )(2 4227 baba
a. 692 ba b. 8142 ba c. 652 ba d. 65ba_______
2. What is the slope of a line passing through the points (5,-2) and (2,-5)?
a. 1 b. -1 c. 37
d. 73
_______
3. If the equation 1836 yx is shifted up 5 units, what is the new y-intercept of the graph?
a. 1 b. -13 c. -1 d. -23_______
4. Solve the following inequality: 3593 xx
a. 6x b. 6x c. 6x d. 6x_______
5. What is the 100th term in the following sequence: -11, -2, 7, 16, 25, …
a. 10,000 b. 880 c. 889 d. 920_______
6. A band class is selling tickets to their concert. If they spent $300 preparing the production, and ticketsare sold for $6, which equation below shows the profit (p) made by the group after selling (t) tickets?
a. p=6t+300 b. p=6t-300 c. p=600t+300 d. p=600t-300
_______
7. Which equation graphed below would result in an upside-down parabola?
a. xxy 452 b. xy 5 c. xy 2 d. 2)2( xy
_______
8. Multiply: 2))(( yxyx
a. 33 yx b. 3223 yxyyxx c. 3223 yxyyxx d. 3223 yxyyxx
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #49. What is equation for a vertical line which passes through (-2, -3)?
a. x=-2 b. x+3=0 c. y=-2 d. y+5=0
_______
10. Which equation below has a graph with a slope of ½?
a. x-2y=12 b. y-2x=12 c. y=2x+12 d. y+2x=12_______
11. Which equation below represents the data given in the table?
x 3 5 7 9 11 13f(x) 18 12 6 0 -6 -12
a. 22)( xxf b. 273)( xxf c. 273)( xxf d. 931)( xxf
_______
12. What is the equation for a line passing through (3, 4) parallel to 1223 xy ?
a. )4(323 xy b. )3(
324 xy c. )3(
324 xy d. )3(
324 xy
_______
13. If the equation 23 xy were graphed, which of the four quadrants would be shaded completely?
a. 1st b. 2nd c. 3rd d. 4th_______
14. What is the range for the function 2)( 2 xxf ?
a. }2{ y b. }2{ y c. }2{ y d. {all real numbers}
_______
Name________________________ Period _____
EOC REVIEW26 Questions EOC Review #4Name________________________ Period _____
15. Which equation below could be used to find the roots of: 11102 xxy
a. )1)(11(0 xx b. )10(0 xx c. )1)(11(0 xx d. )1)(11(0 xx
_______
16. In the sequence below, the 25th term is 83,886,080. What is the 26th term?
5, -10, 20, -40, 80, …
a. 125,829,120 b. -125,829,120 c. 167,772,160 d. -167,772,160_______
17. Write an equation based on the table below showing the amount earned a babysitting based on thenumber of hours worked h, including an initial fee.
hours worked h 3 5 7 9 11earned amt. a $21 $33 $45 $57 $69
a. 5.45.5 ha b. 36 ha c. ha 7 d. 65 ha_______
18. In problem number 17 above, what does the slope represent?
a. Hours worked b. Total Earnings c. Initial fee d. Hourly wages_______
19. Which set of points below does NOT represent a function?
a. (-2, 3) (-3, 4) (-4, 5) (-5, 6)
b. (-2, -2) (-3, -3) (-4, -4) (-5, -5)
c. (-2, 2) (-3, 3) (-2, 4) (-3, 5)
d. (-2, -3) (-3, -2) (-5, -4) (-4, -5)_______
20. The height of a flare fired from a gun can be described by: tth 24016 2 where t is the time inseconds and h is the height in feet. How long will it take for the flare to reach 900 feet?
a. 4 seconds b. 5 seconds c. 6 seconds d. 7 seconds
_______
EOC REVIEW26 Questions EOC Review #4Name________________________ Period _____
21. Solve for x: 023 2 xx
a. }1,7.{ xx b. }1,5.1{ xx c. }1,7.{ xx d. no solutions_______
22. A rectangle is three inches longer than twice its width. Its perimeter is 36 inches. How long is therectangle?
a. 5 inches b. 8 inches c. 13 inches d. 15 inches
_______
23. What would be the new equation for 92 yx if it were shifted up 5 units?
a. 42 xyb. 42 xy
c. 1321
xy
d. 132 xy_______
24. About how many years will it take $9,000 invested at 14% annual compound interestto double in value?
trpA )1(
a. 2.5 years b. 5.3 years c. 7.1 years d. 9.0 years_______
25. Solve the following system of equations: 92 yx332 yx
a. (-3, -6) b. (6, -3) c. (3, 6) d. (-3, 6)
_______
26. For
2314
a and
7586
b find ab .
a.
9892
b.
9892
c.
98
92d.
9892
_______
EOC REVIEWYou must be able to work with polynomials on the EOC:
Distribution and FOIL:Rewrite each using distribution:
1. )3(2 xx 2. )3(5 xx 3. )3)(5( xx
Practice:Distribute. Simplify where possible.
1. )7(5 xxx 2. )5)(72( xx 3. )35)(35( xxPractice:Special Products.
1. )7)(7( xx 2. 2)72( x 3.
2)3( yx Factoring: GCF.Rewrite each by factoring the GCF:
1. xx 302 2 2. 2552 69 yxyx 3. 164 2 x
Factoring: Easy Ones.Example: Factor.
1. 3532 xxPractice: Easy Ones.Factor each.
1. 542 xx 2. 862 xx 3. 72 xx
Polynomials, FOIL, and Factoring
EOC REVIEWFactoring: Special Products.Example: Factor.
1. 49429 2 xx 2. 649 2 xPractice: Special Products.Factor each.
1. 25204 2 xx 2. 1121 2 x
Factoring: Hard Ones (Magic Number).Example: Factor.
1. 102 2 xx 2. 21115 2 xxPractice: Magic Number.Factor each.
1. 927 2 xx 2. 5136 2 xx
Factoring: Solving a quadratic by factoring.Example: Solve for x.
1. 0)12)(3( xx 2. 030112 xxPractice: Solve by factoring.Solve for x.
1. 1522 xx 2. 259 2 x
Polynomials, FOIL, and Factoring
EOC REVIEWDirect Variation is just Slope-Intercept Form (without the intercept).
If a problem states: y varies directly as x,That means y=kx for some value k.
kxy k is called the constant of variation.
You can think if it as slope.
You can also say that x varies directly as y: this means x=ky
Example 1:When y=6, x=2, solve for x when y=-9 if y varies directly as x.
Example 2:The distance it takes to stop a moving train varies directly with the speed it istraveling. A train that is moving 50mph requires 10,000 feet to stop. Howmany feet will be required to stop a train moving 45mph?
Practice:Solve each using direct variation. In each problem, y varies directly as x.
1. When y=8, x=5. What is the constant of variation?2. When x=2, y=7. Find y when x=3.3. The skid marks left by a vehicle can be used to determine the speed withwhich it was traveling. If an 18-wheeler leaves 200-foot skid marks, it wastraveling approximately 60mph. How fast was an 18-wheeler traveling if it left240-foot skid marks?
Proportional Reasoning can also be used for most of these problems:Example 1:The mass of an element varies directly as its volume. If 10cm3 of Carbonweighs 22.6 grams, how much will 12cm3 weigh?a. Solve using a proportion.b. Solve using diect variation.c. What is the constant of variation?d. What do we call the constant of variation in this problem?
Practice:The amount of stretch of a rubber band varies directly as the force applied toit. If a 10-gram weight stretches a rubber band by 8cm, how much will a rub-ber band stretch when weighted with a 14-gram weight?
Direct Variation
EOC REVIEWThe distance between coordinates on the plane can be found usingthe pythagorean theorem.
212
212 )()( yyxxd
The midpoint between two points (x1 , y1) and (x2 , y2) is found byaveraging the x and y coordinates.
Midpoint
2
,2
2121 yyxx
Example: Find the midpoint and distance between (4,3) and (-2, 5).
Practice: Find the midpoint and distance for each pair of points. Leave thedistances in radical form.
1. (8, -2) and (4, -5) 2. (-3, -6) and (7, 3) 3. (8, 9) and (3, 1)
Parallel lines have the same slope.Perpendicular lines have negative reciprocal slopes.
Example: Which lines are parallel? Which are perendicular?
a. 732 yx b. 732 yx c. 723 yx d. 732
xy
Example: The points A(0,7) B(5,8) C(8,2) and D(3,1) form a quadrilateral.Is quadrilateral ABCD a parallelogram?
Practice:1. What is the equation of a line perpendicular to 75 yx through the
point (9,1) in Standard Form?
2. Points A(4,3) B(-8,1) and C(5,-3) are graphed on the plane to form a righttriangle. Which vertex is the right angle of triangle ABC?
3. What is the approximate perimeter of triangle ABC above (to the tenth)?
Distance/Midpoint Parallel/Perp.
EOC REVIEWIf an equation is linear, it has a constant slope.Many EOC problems will ask you to find an answer using a linearmodel.
Examples:In 1970, the average life expectancy in the U.S was 75.2 years. In 2000, theaverage life expectancy was 78.8 years. Assuming the trend is linear, what willbe the average life expectancy in 2020?
The oak tree in your backyard is 15 feet tall. When you planted it 2 years ago,it was just 7 feet tall. If the growth can be modeled by a linear equation, howtall will the tree be in 5 years?
PracticeAssume all growth or depreciation is linear for the following:
1. In the year 2000, a 40-inch LCD television cost about $2,500. In 2008, youcan buy the same television for about $1,100. Assuming a linear rate of depre-ciation, how much less does the television cost this year than last year?
2. Attendance at Lincoln high school has increased linearly for the past 10years. 5 years ago, Lincoln had 1,235 students. Now Lincoln has 1,705 stu-dents. If the growth continues, how many students will attend Lincoln in 3years?
3. The height of a burning candle can be expressed by a linear equationwhere h is the height and m is the number of minutes the candle has beenburning. If a 15-inch candle burns for 35 minutes, it will be 8 inches tall. Whatequation can be written for the height h of a 15-inch candle that has beenburning for m minutes?
Practice: Write a linear equation in slope-intercept form for each:
1. A restaurant takes 1 hour to prepare and bake 16 pizzas. To prepare andbake 20 pizzas takes the same restaurant takes 76 minutes. Whatequation can be used to represent the time in minutes m to bakep pizzas?
2. The length of a particular snake is 16 inches at age 1 and 30 inchesat age 3. Assuming the snake grows at a linear rate, how long was itat birth?
Linear Regression
EOC REVIEWExponential growth:
trpV )1( Amount principal rate (percent as a decimal) time (usually in years)
Examples:How much will $400 be worth in 5 years at 7% interest (to the cent)?
How long will it take for your money to double earning 6% interest?A. 1.5 years B. 10 Years C. 12 Years D. 120 years
Practice:1. The value of an automobile depreciates exponentially. If the rate of depre-ciation on your 7 year-old car is 20%, and the purchase price was $16,000,approximately how much is the car worth today?
A. $18,380 B. $13,890 C. $3,748 D. $3,355
2. Approximately what interest rate must you earn on an investment to doublethe value of your money every 15 years?
A. 4% B. 5% C. 6% D. 7%
3. In 2001, the population of Poughkeepsie was 12,500. In 2007, the popula-tion had increased by 1,170. Assuming that the growth rate is exponential,what is the annual rate of growth?
A. 1.2% B. 1.5% C. 9.4% D. 15.0%
Practice:1. How long will it take an investment to triple in value if it is earning 4.2%interest annually?
A. 17 years B. 20 years C. 23 years D. 26 years
2. Allentown has a population of 50,000 and a 1.5% annual population growthrate. Brighton has 40,000 and a growth rate of 1.8%. If these growth ratescontinue, approximately how many years will it take for the populations to beequal?
A. 25 years B. 50 years C. 75 years D. 100 years
3. An aggressive species of vine grows in length by 20% daily. If the vine is 8inches long when planted, about how long will it be after two weeks of growth?
A. 4 feet B. 5.5 feet C. 7 feet D. 8.5 feet
Exponential Growth/Depreciation
EOC REVIEW
The vertex: abx
2
To find the y-coordinate, plug-in x.
The Roots (solutions, zeros) a
acbbx2
42
Examples:Find the roots and vertex for the following quadratic equation:
344 2 xxy
Solve for x: 10133 2 xx
A.
5,
32
B.
5,
32
C.
5,
32
D.
5,
32
Practice:1. The formula for the height of a ball h after t seconds is given
by the formula: 46416 2 tthWhat is the maximum height of the ball?
A. 4 feet B. 60 feet C. 64 feet D. 68 feet
2. Find the roots: 15164 2 xxh
A.
25,
23
B.
25,
23
C.
25,
23
D.
25,
23
3. The length of a rectangle is 4 inches greater than twice the length.If the rectangle has an area of 70in2, what is its perimeter?A. 34in B. 38in C. 40in D. 74in
Quadratics
EOC REVIEWExamples:
Where would the graphs of the two equations below intersect?
A. 34 xy B. 923 yx
Write and solve a system of equations to solve the following:At a local bakery, 3 pastries and 5 doughnuts cost $6.75.At the same bakery, 4 pastries and one doughnut cost $4.75.How much would 2 pastries and 2 doughnuts cost?
A. $3.25 B. $3.50 C. $3.75 D. $4.00
Practice:
1. Solve for x in the following system of equations: 42 yx 853 yx
A. 127
x B. 712
x C. 1328
x D. 2813
x
2. Micah has 4 more pencils than he has erasers. If he has a total of 42 pen-cils and erasers, how many pencils does he have?
A. 19 B. 21 C. 23 D. 25
Practice:
1. The perimeter of a rectangle is 31 inches, and the width is 10 inchesless than twice the length. What is the width of the rectangle?A. 7in B. 7.5in C. 8in D. 8.5in
2. Adult tickets to a theme park cost $19 and childrens tickets cost $15.One cashier collected $632 from 40 park attendants.How many adult attendants paid the cashier?A. 8 B. 16 C. 24 D. 32
Systems of Equations
EOC REVIEWFormulas you must know:
Fill-in the following formulas. These are all formulas that you must be able to use on the EOCtest Tuesday. You will be quizzed on these formulas Wednesday so memorize these!
1. Point-Slope Form: _____________________________________________
2. Slope-Intercept Form: ___________________________________________
3. Standard Form: _________________________________________________
4. Slope (given two points): ___________________________________________
5. Distance Formula (given two points): _________________________________
6. Midpoint Formula (given two points): _________________________________
7. Quadratic Formula: _____________________________________________
8. X-coordinate of the vertex of a quadratic: ______________________________
9. Slope of a standard form linear equation: __________________________
10. Direct Variation: ___________________________________________________
11. Exponential Growth: ______________________________________________
Go to
http://www.ncpublicschools.org/accountability/testing/eoc/sampleitems/alg1scs2003extset
for sample items if you are looking for something to review tonight but GET AGOOD NIGHT’S REST!
Name________________________ Period _____
Formulas
North Carolina Testing Program EOC Algebra I Sample Items Goal 1
Page 1 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. What is the greatest common factor of 4 3 3 3 2 215 21 6 ?x y x y x y− +
A 2 2x y
B xy
C 23xy
D 2 23x y
2. Simplify: 3 2 2 3
214 21
14c d c d
cd-
A 2 3
2cdc -
B 2
2 32c dc -
C 2 2 221c c d-
D 2 32cdc d -
3. 2Simplify: ( 2)( 2 3)x x x+ + +
A 3 7 6x x+ +
B 25 7 6x x+ +
C 3 22 6x x x+ + +
D 3 24 7 6x x x+ + +
4. Which binomial is a factor of 23 2 5?x x+ -
A 3 1x -
B 1x -
C 3 5x -
D 5x -
5. What is the quotient when
( )4 26 9 12x x x- + is divided by 3 ?x
A 4 22 3 4x x x- +
B 32 6 4x x+ +
C 32 3 4x x+ +
D 32 3 4x x- +
6. Which expression is a factor of
( )3 26 13 28 ?x x x- -
A 4x −
B 2 7x −
C 2 7x +
D 3 4x −
North Carolina Testing Program EOC Algebra I Sample Items Goal 1
Page 2 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
7. Multiply:
( ) ( )—1 26 4 5 33 3x y x y
A 11 7—27x y
B 11 7—18x y
C 4 22x y
D 4 23x y
8. Suppose that the value, V, of a used
machine can be calculated by using
the formula ( )1 ,20nV P= − where P
represents the price of a new machine
and n represents the machine’s age
in years. A company purchased a new
machine for $15,000. The value of the
machine is now $12,375. How old is
the machine?
A 1.2 years
B 3.5 years
C 4.3 years
D 5.7 years
9. To find the image length, L, of a
4-foot-tall object in a spherical
mirror with a focal length of 2 feet,
( )224 2L o= − can be used, where
o is the distance, in feet, of the
object from the mirror. What is the
image length of the object when it
is 1.5 feet away from the mirror?
A 256 feet
B 128 feet
C 64 feet
D 32 feet
North Carolina Testing Program EOC Algebra I Sample Items Goal 1
Page 3 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
10. The following figures are created with regular pentagons. Each pentagon has a side
length of one unit. 1P is the perimeter of the first figure, 2P is the perimeter of the
second figure, and so on.
P1 = 5 P2 = 8 P3 = 11n = 1 n = 2 n = 3
P4 = 14n = 4
According to this pattern, what would be the rule for the perimeter, ,nP of the nth figure when 1?n >
A 12 2n nP P -= -
B 12 4n nP P -= +
C 1 3n nP P -= -
D 1 3n nP P -= +
North Carolina Testing Program EOC Algebra I Sample Items Goal 1
Page 4 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
11. Which expresses the total surface area (including the top and bottom) of a tower of c cubes each having side length e? (do not include faces that cover each other)
c = 3c = 2c = 1
A ( ) 24 2c e+
B 3c e∑
C 26c e∑
D 24c e∑
12. The number of bacteria in an
experiment can be represented by
the formula 1 2.5 .t tN N+ = In the
formula, tN is the number of
bacteria at the end of t minutes,
and 1tN + is the number of bacteria
at the end of 1t + minutes. There
are 16,400 bacteria in the
experiment at the end of 7 minutes.
How many bacteria will be in the
experiment at the end of 10 minutes?
A 23,429
B 102,500
C 123,000
D 256,250
13. Hooke’s law states that the distance a vertical spring stretches varies directly with the weight hanging from it. A spring stretches 14 inches when a 35-pound weight is hanging from it. How much weight is needed to stretch the spring 44 inches?
A 110 pounds
B 65 pounds
C 17.6 pounds
D 11.1 pounds
North Carolina Testing Program EOC Algebra I Sample Items Goal 1
Page 5 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
14. When x is 3, y is 12. If y varies directly as x, which equation relates x and y?
A 9y x= +
B 15y x= −
C 36y x=
D 4y x=
15. Suppose that y varies directly as x,
and y = 5 when x = 2. What is the value of y when x = 7?
A 2.8
B 10
C 17.5
D 35
16. Neglecting reaction time, the distance required for a car to stop is directly proportional to the square of its velocity. If a car can stop in 8.5 meters at 20 kilometers per hour,
how approximately many meters are needed to stop at 50 kilometers per hour?
A 13.4
B 21.3
C 53.1
D 117.6
End of Goal 1 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
EOC Algebra 1 Goal 1 Sample Items Key Report
Page 1
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: D 2 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: A 3 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: D 4 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: B 5 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: D 6 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: B 7 Objective: 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of
exponents. b) Operate with polynomials. c) Factor polynomials. Thinking Skill: Applying Correct Answer: D 8 Objective: 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to
model and solve problems. Thinking Skill: Analyzing Correct Answer: B
EOC Algebra 1 Goal 1 Sample Items Key Report
Page 2
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
9 Objective: 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to
model and solve problems. Thinking Skill: Applying Correct Answer: C 10 Objective: 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to
model and solve problems. Thinking Skill: Analyzing Correct Answer: D 11 Objective: 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to
model and solve problems. Thinking Skill: Analyzing Correct Answer: A 12 Objective: 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to
model and solve problems. Thinking Skill: Applying Correct Answer: D 13 Objective: 1.03 Model and solve problems using direct variation. Thinking Skill: Applying Correct Answer: A 14 Objective: 1.03 Model and solve problems using direct variation. Thinking Skill: Applying Correct Answer: D 15 Objective: 1.03 Model and solve problems using direct variation. Thinking Skill: Applying Correct Answer: C 16 Objective: 1.03 Model and solve problems using direct variation. Thinking Skill: Applying Correct Answer: C
North Carolina Testing Program EOC Algebra I Sample Items Goal 2
Page 1 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. What is the area of a square with vertices (3, 3), (6, 6), (9, 3), and (6, 0)?
A 23 2 units
B 212 2 units
C 218 units
D 236 units
2. On a map’s coordinate grid,
Panthersville is located at ( )—3, 2 ,
and Heel City is located at ( )4, 8 .
Falconton is the midpoint between
Panthersville and Heel City. What
is the approximate distance from
Panthersville to Falconton?
(One map unit equals one mile.)
A 3.25 miles
B 4.61 miles
C 5.00 miles
D 9.22 miles
North Carolina Testing Program EOC Algebra I Sample Items Goal 2
Page 2 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
3. What is the perimeter of ?PQR
x
y
Q (—2, 3)
R (3, —2)
P (0, —5)
A 136
B 10 21
C 2 5 2 3 17 2+ +
D 8 2 2 17+
North Carolina Testing Program EOC Algebra I Sample Items Goal 2
Page 3 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
4. Given points ( ) ( ) ( )—7, 5 , 8, 3 , 0, 1 ,P Q R
and ( )—1, 1 ,S which statement is true?
A is parallel to .PQ RS
B is perpendicular to .PQ RS
C is perpendicular to .PR QS
D is parallel to .PR QS
5. Line segment RS is perpendicular
to line segment PQ, and the
coordinates are ( )—4, 5 ,R ( )—8, 4 ,S
( )0, 6 ,P and ( )—3, .Q y What is the
value of y?
A 9
B 8.25
C 2
D 23
6. The equation of the line containing
one side of a parallelogram
is 3 2 8.x y+ = The opposite side
contains the point ( )—0, 7 . Which is
the equation of the line that contains
the opposite side?
A 23
7y x= −
B 32
7y x−
= +
C 23
7y x= +
D 32
7y x−
= −
7. Which of the following is an
equation of the line perpendicular to 3 6 12 and passing through (4,0)?
x y+ =
A —1
2 2y x= +
B 12 2y x= −
C —2 8y x= +
D 2 8y x= −
North Carolina Testing Program EOC Algebra I Sample Items Goal 2
Page 4 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
8. The line passing through points
( ), 4x and ( )—4, 5 is perpendicular
to a line with a slope of 73
.−
What is
the value of ?x
A —17
B 17
C 557
D 25
End of Goal 2 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
EOC Algebra 1 Goal 2 Sample Items Key Report
Page 1
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 2.01 Find the lengths and midpoints of segments to solve problems. Thinking Skill: Analyzing Correct Answer: C 2 Objective: 2.01 Find the lengths and midpoints of segments to solve problems. Thinking Skill: Analyzing Correct Answer: B 3 Objective: 2.01 Find the lengths and midpoints of segments to solve problems. Thinking Skill: Applying Correct Answer: D 4 Objective: 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems. Thinking Skill: Applying Correct Answer: A 5 Objective: 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems. Thinking Skill: Analyzing Correct Answer: C 6 Objective: 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems. Thinking Skill: Analyzing Correct Answer: D 7 Objective: 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems. Thinking Skill: Analyzing Correct Answer: D 8 Objective: 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems. Thinking Skill: Analyzing Correct Answer: D
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 1 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. Which matrix contains the coordinates of the parallelogram shown below?
—9 —8 —7 —6 —5 —4 —3 —2 —1—10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +100—1—2—3—4—5—6—7—8—9
—10
+10+9+8+7+6+5+4+3+2+1
x
y
(10, 7)(4, 7)
(8, 3)(2, 3)
A ⎡ ⎤⎢ ⎥⎣ ⎦2 3 3 47 7 8 10
B 2 4 10 87 3 3 7⎡ ⎤⎢ ⎥⎣ ⎦
C 2 4 7 33 7 10 8⎡ ⎤⎢ ⎥⎣ ⎦
D 2 4 10 83 7 7 3⎡ ⎤⎢ ⎥⎣ ⎦
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 2 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
2. The Baltic Sea covers 147,500 square miles of area and has an average depth of 180 feet. The North Sea covers 164,900 square miles of area and has an average depth of 308 feet. The Red Sea has an area of 174,900 square miles and has an average depth of 1,764 feet. The East China Sea has an area of 256,600 square miles and an average depth of 620 feet. Which matrix displays this information organized by area and depth of each sea?
A 147,500 164,900 174,900 256,600180 308 620 1,764
⎡ ⎤⎢ ⎥⎣ ⎦
B 0
10
47,500 180 164,900 308174,900 1 000,764 256,600 6, 20⎡ ⎤⎢ ⎥⎣ ⎦
C 147,500 164,900 174,900 256,600180 308 1,764 620
⎡ ⎤⎢ ⎥⎣ ⎦
D 147,500 164,900 174,900 256,600180 620 308 1,764
⎡ ⎤⎢ ⎥⎣ ⎦
3. This matrix shows the cost of cell phone service offered by several different companies.
$39.00
$27.00
$42.00
$30.00
Monthly Costfor 200 Minutes
Company 1
Company 2
Company 3
Company 4
$0.05
$0.08
$0.04
$0.06
Cost of Each Minuteover 200 Minutes
What is the cost of 320 minutes with Company 4?
A $37.20
B $45.00
C $49.20
D $75.00
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 3 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
4. On Tuesday, a store sold 12 compact discs, 5 cassettes, and 9 videos. On Wednesday, the store sold 19 compact discs, 3 cassettes, 9 videos, and 35 concert tickets. Which matrix shows the number of items sold, organized by day and product?
A 12 9 519 3 9È ˘Í ˙Î ˚
B 12 5 919 3 35È ˘Í ˙Î ˚
C 12 5 9 019 3 9 35È ˘Í ˙Î ˚
D 12 9 19 95 0 3 35
È ˘Í ˙Î ˚
5. The matrix below shows the cost of a school lunch at four schools over a four-year period.
1.50
1.45
1.40
1.40
Year 1School 1
School 2
School 3
School 4
1.50
1.46
1.60
1.42
Year 21.60
1.50
1.60
1.45
Year 31.75
1.72
1.62
1.65
Year 4
Which school had the greatest increase in the cost of a school lunch over the four-year period?
A School 1
B School 2
C School 3
D School 4
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 4 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
6. The matrix below displays the average SAT scores of eleventh- and twelfth-grade students over a three-year period at a high school.
1998
976
1,028
1999
1,035
1,164
2000
1,100
1,253
Grade 11
Grade 12
Carter High SchoolAverage SAT Scores
What was the change in average SAT scores of the twelfth-graders from 1998 to 2000?
A Scores increased by 225 points.
B Scores increased by 89 points.
C Scores decreased by 225 points.
D Scores decreased by 89 points.
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 5 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
7. A survey was done asking students what type of athletic shoes they wear and which type they would buy the next time they bought shoes. The results are shown in the chart below.
Type ofShoe Worn
Type of ShoeStudents Would Buy
Tennis Shoes (T)
Running Shoes (R)
40% Tennis Shoes (T)25% Running Shoes (R)35% Basketball Shoes (B)
60% Running Shoes (R)15% Tennis Shoes (T)25% Basketball Shoes (B)
Which matrix represents these data?
A
TR
40%
15%
T
35%
25%
B
25%
60%
R
B
TR
40%
60%
T
25%
15%
B
35%
25%
R
C
TBR
40%
25%
35%
T
60%
15%
25%
R
D
TBR
40%
35%
15%
T
25%
60%
25%
R
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 6 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
8. Matrices P and Q are shown below.
P = 3 26 91 0
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
Q =
— —
— —3 72 64 0
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
What is P Q− ?
A — —
— —6 98 153 0
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
B —0 54 35 0
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
C — —
—
0 54 35 0
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
D
—
6 98 153 0
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 7 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
9. Given the matrices:
26 18 43 2134 19 and 26 2061 23 33 92
J K⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
What is 4 2 ?J K−
A — —
— —
— —
120 4836 4210 322
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
B
—
18 3084 36
178 92
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
C 120 4836 4210 322
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
D — —
—
—
17 38 1
28 69
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 8 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
10. Given the matrices:
—
— — —3 2 1 5 4 07 2 3 2 6 1
E F⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
What is 3 ?E F−
A —
4 10 319 0 10⎡ ⎤⎢ ⎥⎣ ⎦
B —
—6 18 3
15 12 12⎡ ⎤⎢ ⎥⎣ ⎦
C — —
4 2 019 12 8⎡ ⎤⎢ ⎥⎣ ⎦
D —
—2 6 15 4 4
⎡ ⎤⎢ ⎥⎣ ⎦
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 9 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
11. Given:
— —
—
———
1 2 5 1 4 30 5 9 10 2 , , and8 0 4 76 18 1 3 45 9
J K L
⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥= = = ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
What is ?J K L+ −
A —
—
—
8 09 62 8
16 12
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
B —
— —
—
0 69 86 6
10 4
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
C —
—
2 29 4
18 80 14
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
D —
—
—
10 49 2
10 66 6
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 10 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
12. What is
—2 3 2 64 33 4 3 7
⎡ ⎤⎡ ⎤ − ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦?
A 0 96 11⎡ ⎤⎢ ⎥⎣ ⎦
B —
—2 63 5⎡ ⎤⎢ ⎥⎣ ⎦
C 12 303721
⎡ ⎤⎢ ⎥⎣ ⎦
D —
—14 6
3 5⎡ ⎤⎢ ⎥⎣ ⎦
13. Nagel’s Bagel Shop makes a monthly report to summarize the cost of making a single
bagel of each type and the price at which it is sold. Matrix C represents cost, and matrix Srepresents selling price.
[ ] [ ]Plain Blueberry Wheat Onion Plain Blueberry Wheat Onion0.12 0.17 0.13 0.15 0.45 0.50 0.50 0.50C S= =
Which matrix represents the profit on the sale of a single bagel of each type? (Profit = Selling Price — Cost)
A [ ] Plain Blueberry Wheat Onion
0.57 0.67 0.63 0.65
B [ ] Plain Blueberry Wheat Onion
0.33 0.33 0.35 0.37
C [ ] Plain Blueberry Wheat Onion
0.33 0.33 0.33 0.33
D [ ] Plain Blueberry Wheat Onion
0.33 0.33 0.37 0.35
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 11 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
14. The table shows the relationship between calories and fat grams contained in orders of fried chicken from various restaurants.
Calories 305 410 320 500 510 440
Fat Grams 28 34 28 41 42 38
Assuming the data can best be described by a linear model, how many fat grams wouldbe expected to be contained in a 275-calorie order of fried chicken?
A 28
B 27
C 25
D 22
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 12 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
15. The equation 0.117 39,905y x= +
models the relationship where x is the total population of a state, and y is
the number of people aged 65 years
or older in a state. Suppose the
difference in total population between
two states is one million. According
to the equation, what is the expected
difference in the number of people
aged 65 or older?
A 39,905
B 117,000
C 0.117 (39,905)
D 39,905 0.117÷
16. Five students in Miss Brown’s algebra class reported the number of hours that they studied for a test. The number of hours and their test scores are in the table below.
Hours of Study Test Score
86
80
85
90
96
2
2.5
3
4.5
5 According to a line of best fit for the data, what is the predicted test score of a student who studied 1 hour for the test?
A 75
B 78
C 81
D 84
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 13 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
17. The Smiths’ average monthly electric bills in the years 1998 to 2005 are displayed in the table below.
Year
Average Monthly Bill
1998
$102
1999
$102
2000
$104
2001
$108
2002
$116
2003
$116
2004
$121
2005
$129
According to a line of best fit for the data, approximately how much per month would theSmiths pay in 2007?
A $134
B $137
C $142
D $145
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 14 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
18. ( )The graph shows a scatterplot of the number of compact discs CDs sold at a music store during part of the 1980s and early 1990s. An equation for the line of best fit for the given data is 518 43,886y x= − .
908886
1,5002,0002,5003,000
x
y
5,0004,5004,0003,500
1,000500
84 92 94Year
(88, 1,497)
What is the difference between the observed value and the predicted value at 88?x =
A 1,698
B 979
C 518
D 201
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 15 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
19. The chart below shows cell phone use for seven years.
Year
1999200020012002200320042005
Number ofCell Phone Minutes
(billions)
156264401549705857
1,000
According to the line of best fit for the data, what is the approximate average annual increase in cell phone minutes for 1999—2005?
A 108 billion minutes
B 121 billion minutes
C 141 billion minutes
D 144 billion minutes
20. The table below shows the price of rings for various weights of gemstones.
Weight (x)
Price (y)
0.17
$355
0.25
$642
0.28
$823
0.35
$1,086
0.32
$919 Which statement best interprets the meaning of the y-intercept of the linear function that best fits these data?
A the price of the ring per unit of weight of the gemstone
B the weight of the gemstone per dollar
C the cost of the ring with no gemstone
D the weight of the gemstone in the ring that costs $0
North Carolina Testing Program EOC Algebra I Sample Items Goal 3
Page 16 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
21. The table below shows the number of doctors in Bingham City from 1960 to 1986.
1960
2,937
1967
3,511
1970
3,754
1975
4,173
1982
4,741
1985
5,019
1986
5,102
Year(x)
Number ofDoctors
(y)
If a linear regression model is fit to this data, which equation would represent the data? (let the number of years after 1960)x =
best
A 1.01 3,500y x= −
B 82 2,937y x= +
C 83 2,929y x= +
D 83 2,944y x= +
End of Goal 3 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
EOC Algebra 1 Goal 3 Sample Items Key Report
Page 1
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Organizing Correct Answer: D 2 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Organizing Correct Answer: C 3 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Analyzing Correct Answer: A 4 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Organizing Correct Answer: C 5 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Applying Correct Answer: B 6 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Applying Correct Answer: A 7 Objective: 3.01 Use matrices to display and interpret data. Thinking Skill: Organizing Correct Answer: A 8 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: D 9 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: B 10 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: A
EOC Algebra 1 Goal 3 Sample Items Key Report
Page 2
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
11 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: D 12 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: D 13 Objective: 3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems. Thinking Skill: Applying Correct Answer: D 14 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: C 15 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: B 16 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: B 17 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: A 18 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: D
EOC Algebra 1 Goal 3 Sample Items Key Report
Page 3
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
19 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: D 20 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Analyzing Correct Answer: C 21 Objective: 3.03 Create linear models for sets of data to solve problems. a) Interpret constants and
coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
Thinking Skill: Applying Correct Answer: C
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 1 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. Denisha bought a car for $15,000 and its value depreciated linearly. After 3 years the value was $11,250. What is the amount of yearly depreciation?
A $2,000
B $1,500
C $1,250
D $750
2. In 1977, the price of a scientific
calculator was $175. In 2004, the price was $15. Assuming the change in price was linear, what was the approximate price of a scientific calculator in 1998?
A $23.00
B $27.00
C $51.00
D $60.00
3. The attendance on the first day of a carnival was 425 people. The attendance on the third day was 575 people. Assuming attendance will increase linearly each day, how many people will attend the carnival on the sixth day?
A 650
B 725
C 800
D 875
4. Jim is selling hot dogs at a ball game. It cost Jim $250 to purchase everything to make the hot dogs. Jim sells hot dogs for $2.00 each. If he sells h hot dogs, which equation models his profit (P)?
A 2 250P h= +
B 2 250P h= −
C 250 2P h= +
D 250 2P h= −
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 2 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
5. According to the graph, which statement best describes the slope?
10
5
10050Distance Traveled
(miles)
A As the distance traveled increases by 20, the amount of gas in the tank decreases by 3.
B As the distance traveled decreases by 3, the amount of gas in the tank increases by 20.
C As the distance traveled increases by 30, the amount of gas in the tank increases by 2.
D As the distance traveled decreases by 20, the amount of gas in the tank decreases by 3.
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 3 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
6. If the graph of a line has a positive slope and a negative y-intercept, what happens to the x-intercept if the slope and the y-intercept are doubled?
A The x-intercept becomes four times larger.
B The x-intercept becomes twice as large.
C The x-intercept becomes one-fourth as large.
D The x-intercept remains the same.
7. Nancy earns $200 per week plus
15% commission on the value of her sales. In the linear function representing Nancy’s weekly earnings, x represents the value of her sales, and y represents her total earnings for the week. What does the y-intercept of the function represent?
A the amount of commission earned for one week
B the rate of commission on the value of her sales
C the total earnings for one week when she makes $0 in sales
D the value of her sales for one week when she makes $0 in total earnings
8. In 1994, the average price of a new domestic car was $16,930. In 2002, the average price was $19,126. Based on a linear model, what would be the approximate predicted average price for 2008?
A $23,000
B $21,300
C $20,800
D $18,600
9. The cost of renting a van for one day includes a flat rental fee plus a charge for each mile the van is driven while it is rented. A van that is driven 107 miles costs $97.15. A van that is driven 127 miles costs $106.15. What is the flat rental fee?
A $19.00
B $20.00
C $45.00
D $49.00
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 4 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
10. ( )( )
02
0
An object is blasted upward at an initial velocity, , of 240 ft/s. The height, , of the object is a function of time, (in seconds), and is given by the formula 16 . How long will it tak
v h tt h t v t t= −
e the object to hit the ground after takeoff ?
A 16 seconds
B 15 seconds
C 7.5 seconds
D 4 seconds
11. The area of the rectangle shown in the diagram below is 2170 ft .
3x + 2
2x
What is the perimeter of the rectangle?
A 27 ft
B 40 ft
C 54 ft
D 68 ft
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 5 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
12. ( ) 2—Given 3 5, what isthe range of the function?
f x x= +
A all real numbers less than or equal to 5
B all integers less than or equal to 5
C all nonnegative real numbers
D all nonnegative integers
13. What are the roots of 20 9 49?x= −
A 7±
B 3±
C 49
9±
D 73
±
14. Tim kicks a ball off the ground. After
t seconds, its height, h (in feet), is given by the formula 2—16 64 .h t t= + What is the maximum height reached by the ball?
A 80 feet
B 64 feet
C 48 feet
D 16 feet
15. What are the approximate solutions of the equation 2 —4 2?x x+ =
A { }—4.45, 0.45
B { }—3.41, 0.45
C { }— —0.59, 4.45
D { }— —0.59, 3.41
16. A store received $823 from the sale of
5 tape recorders and 7 radios. If the receipts from the tape recorders exceeded the receipts from the radios by $137, what was the cost of a tape recorder?
A $49
B $68
C $84
D $96
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 6 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
17. A region is defined by this system:
2 12
y x—y x
> +£ -
In which quadrants of the coordinate plane is the region located?
A I, II, III only
B II, III only
C III, IV only
D I, II, III, IV
18. Given:
—6 3 424 2 4
x yx y
− =+ =
What is ?x y+
A —6
B —5
C 4
D 9
19. Given the system of equations below:
3 2 124 11x yx y− =
− =
What is the value of y in the solution?
A —3
B —2
C 2
D 3
20. A local kennel has twice as many cats (c) as dogs (d). When full, the kennel has a total of 30 cats and dogs. Which system of equations could be used to find the number of cats in the kennel when it is full?
A 230
c dc d=+ =
B 230
d cc d=+ =
C 302
c dc d+ == +
D 230
c dc d= -+ =
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 7 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
21. The junior class sold 120 turkey dinner plates and 200 chicken dinner plates for a total of $2,150. The senior class sold 100 turkey plates and 300 chicken plates, raising $2,625. What was the cost of each turkey dinner plate?
A $6
B $6.25
C $7
D $7.50
22. A city’s population, P (in thousands),
can be modeled by the equation
( )130 1.03 ,x
P = where x is the number
of years after January 1, 2000. For
what value of x does the model predict
that the population of the city will be
approximately 170,000?
A 8
B 9
C 10
D 11
23. A new automobile is purchased for $20,000. If ( )20 000 0 8 xV , .= gives the car’s value after x years, about how long will it take for the car to be worth half its purchase price?
A 3 years
B 4 years
C 5 years
D 6 years
24. The value of Mr. Dulaney’s car x yearsafter its purchase is given by the function ( )( ) 15,000 0.87 .
xV x =
Approximately, what was the value of Mr. Dulaney’s car 5 years
after its purchase?
A $7,500
B $8,600
C $9,900
D $13,100
North Carolina Testing Program EOC Algebra I Sample Items Goal 4
Page 8 Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
25. Three years ago, Andy invested $5,000 in an account that earns 5% interest compounded annually. The equation y = 5,000(1.05)t describes the balance in the account, where t is time in years. Andy made no additional deposits and no withdrawals. How much is in the account now?
A $5,788.13
B $5,750.00
C $5,470.19
D $5,250.26
26. The function ( )58.7 1.03
ty =
gives a country’s population,
y (in millions), where t is the number
of years since January 1994.
According to this function, what
was the approximate population
of the country in January 2002?
A 68 million
B 70 million
C 72 million
D 74 million
27. When Robert was born, his
grandfather invested $1,000 for
Robert’s college education. At an
interest rate of 4.5%, compounded
annually, approximately how
much would Robert have at age 18?
(use the formula ( )1 ,t
A P r= +
where P is the principal, r is the
interest rate, and t is the time in
years)
A $1,810
B $2,200
C $3,680
D $18,810
End of Goal 4 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
EOC Algebra 1 Goal 4 Sample Items Key Report
Page 1
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Applying Correct Answer: C 2 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: C 3 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: C 4 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Organizing Correct Answer: B 5 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: A 6 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: D 7 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: C
EOC Algebra 1 Goal 4 Sample Items Key Report
Page 2
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
8 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Analyzing Correct Answer: C 9 Objective: 4.01 Use linear functions or inequalities to model and solve problems; justify results. a)
Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.
Thinking Skill: Applying Correct Answer: D 10 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Applying Correct Answer: B 11 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Analyzing Correct Answer: C 12 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Analyzing Correct Answer: A 13 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Applying Correct Answer: D 14 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Applying Correct Answer: B 15 Objective: 4.02 Graph, factor, and evaluate quadratic functions to solve problems. Thinking Skill: Applying Correct Answer: D 16 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Analyzing Correct Answer: D
EOC Algebra 1 Goal 4 Sample Items Key Report
Page 3
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
17 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Applying Correct Answer: B 18 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Applying Correct Answer: B 19 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Applying Correct Answer: A 20 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Applying Correct Answer: A 21 Objective: 4.03 Use systems of linear equations or inequalities in two variables to model and solve
problems. Solve using tables, graphs, and algebraic properties; justify results. Thinking Skill: Applying Correct Answer: D 22 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Analyzing Correct Answer: B 23 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Applying Correct Answer: A 24 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Applying Correct Answer: A 25 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Applying Correct Answer: A
EOC Algebra 1 Goal 4 Sample Items Key Report
Page 4
Published December 2007. May reproduce for instructional and educational purposes only; not for personal or financial gain.
26 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Applying Correct Answer: D 27 Objective: 4.04 Graph and evaluate exponential functions to solve problems. Thinking Skill: Applying Correct Answer: B
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 1
Page 1 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
1. A surveyor determined that the distance across a pond is 2,255 feet.
, what is this distance?Approximately
A 22.6 ft
B 25.0 ft
C 47.5 ft
D 1,127.5 ft
2. The area of a square is
800 square meters. The length of its side is between which two numbers?
A 27 m and 28 m
B 28 m and 29 m
C 200 m and 201 m
D 400 m and 401 m
3. Which point represents 2 on the
number line below?
10 3 4
YW ZX
2
A W
B X
C Y
D Z
4. Which number below is an irrational number?
A 23
B 2.35
C 25
D 5
5. Which number has the greatest value?
A 1.5
B 30100
C 19
D 3
6. Which choice lists the three lengths in order from greatest to least?
A 4.5, 3, 18
B 18, 4.5, 3
C 4.5, 18, 3
D 3, 4.5, 18
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 1
Page 2 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
7. Which choice is more than 12 but less than 13?
A 170
B 150
C 144
D 140
8. The area of a triangle is
51 square meters. The height is one-half the length of the base. What is the approximate height of the triangle?
A 7.0 meters
B 7.1 meters
C 14.0 meters
D 14.3 meters
9. The drama club is selling tickets to a play for $10 each. The cost to rent the theater and costumes is $500. In addition, the printers are charging $1 per ticket to print the tickets. How many tickets must the drama club sell to make a profit?
A 54
B 55
C 56
D 57
End of Goal 1 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
Gr08 Math Goal 1 Sample Items Key Report
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Applying Correct Answer: C 2 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Applying Correct Answer: B 3 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Organizing Correct Answer: B 4 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Organizing Correct Answer: D 5 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Organizing Correct Answer: D 6 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Organizing Correct Answer: C 7 Objective: 1.01 Develop number sense for the real numbers. A) Define and use irrational numbers. B)
Compare and order. C) Use estimates of irrational numbers in appropriate situations. Thinking Skill: Organizing Correct Answer: B 8 Objective: 1.02 Develop flexibility in solving problems by selecting strategies and using mental
computation, estimation, calculators or computers, and paper and pencil. Thinking Skill: Evaluating Correct Answer: B
Gr08 Math Goal 1 Sample Items Key Report
Page 2 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
9 Objective: 1.02 Develop flexibility in solving problems by selecting strategies and using mental
computation, estimation, calculators or computers, and paper and pencil. Thinking Skill: Analyzing Correct Answer: C
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 2
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. If the length of a rectangle is doubled, what will happen to its area?
A The area will be the same.
B The area will be twice as large.
C The area will be three times as large.
D The area will be four times as large.
2. The diagram below shows a company’s
current packaging of its plant food.
7 cm
10 cm
The company will double the radius but keep the height the same. What effect will this change have on the volume of the container?
A The new volume will be one and a half times the original volume.
B The new volume will be twice the original volume.
C The new volume will be three times the original volume.
D The new volume will be four times the original volume.
3. A hole shaped like a rectangular prismis 3 feet wide, 5 feet long, and 3 feet deep. If the hole is made 2 feet deeper, by how much will the volume of the hole increase?
A 30 cubic feet
B 75 cubic feet
C 90 cubic feet
D 130 cubic feet
4. A 5 × 7 photo is enlarged so that its new dimensions are 10 × 14. How does the area of the enlarged photo compare to the area of the original photo?
A The area of the enlarged photo is five square units larger than the area of the original photo.
B The area of the enlarged photo is seven square units larger than the area of the original photo.
C The area of the enlarged photo is two times the area of the original photo.
D The area of the enlarged photo is four times the area of the original photo.
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 2
Page 2 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
5. The side measurements of a cube are tripled. What is the ratio of the surface area of the original cube to the surface area of the larger one?
A 1 : 3
B 1 : 6
C 1 : 9
D 1 : 12
6. At noon, the shadow of a flagpole is
19 feet long. At the same time, the shadow of a 12-foot-high wall is 4 feet long. What is the height of the flagpole?
A 48 feet
B 57 feet
C 62 feet
D 75 feet
7. Marissa’s shadow is 8 feet long, and she is 5.5 feet tall. At the same time of day, a building casts a 20-foot shadow. Which proportion canbe used to find the height, x, of the building?
A 5.58 20x =
B 5.520 8x =
C 5.5812
x =
D 125.5 8x =
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 2
Page 3 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
8. Jake wanted to measure the length, l,of the pond, so he drew this diagram oftwo similar triangles.
l
Pond
7 ft
5 ft
18 ft
What is the approximate length, l, of the pond?
A 25 feet
B 19 feet
C 18 feet
D 13 feet
End of Goal 2 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
Gr08 Math Goal 2 Sample Items Key Report
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 2.01 Determine the effect on perimeter, area or volume when one or more dimensions of two-
and three-dimensional figures are changed. Thinking Skill: Applying Correct Answer: B 2 Objective: 2.01 Determine the effect on perimeter, area or volume when one or more dimensions of two-
and three-dimensional figures are changed. Thinking Skill: Analyzing Correct Answer: D 3 Objective: 2.01 Determine the effect on perimeter, area or volume when one or more dimensions of two-
and three-dimensional figures are changed. Thinking Skill: Applying Correct Answer: A 4 Objective: 2.01 Determine the effect on perimeter, area or volume when one or more dimensions of two-
and three-dimensional figures are changed. Thinking Skill: Applying Correct Answer: D 5 Objective: 2.01 Determine the effect on perimeter, area or volume when one or more dimensions of two-
and three-dimensional figures are changed. Thinking Skill: Applying Correct Answer: C 6 Objective: 2.02 Apply and use concepts of indirect measurement. Thinking Skill: Applying Correct Answer: B 7 Objective: 2.02 Apply and use concepts of indirect measurement. Thinking Skill: Applying Correct Answer: B 8 Objective: 2.02 Apply and use concepts of indirect measurement. Thinking Skill: Integrating Correct Answer: D
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 3
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. What is the maximum number of cubes with a side length of 2 inches that can fit in this box?
4 inches
2 inches6 inches
A 48
B 24
C 12
D 6
2. The hypotenuse of a right triangle
measures 20.75 in., and one of the legs measures 17 in. What is the approximate length of the other leg of the triangle?
A 2 in.
B 7 in.
C 12 in.
D 27 in.
3. A ladder leans against the side of a building. The base of the ladder is 5 meters from the building, and the top is 12 meters above the ground.
5 m
12 m
What is the length of the ladder?
A 11 meters
B 13 meters
C 17 meters
D 169 meters
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 3
Page 2 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
4. What is the approximate length of the diagonal of a square with side length of 20 centimeters?
A 14.1 cm
B 20.0 cm
C 25.0 cm
D 28.3 cm
5. Jamie and Fred are meeting Susan at her house. Jamie must use the crosswalk in front of Fred’s house to cross the street.
Susan’sHouse
Fred’sHouse
Jamie’sHouse
112 ft
What is the total distance Jamie must walk?
A 66 ft
B 130 ft
C 178 ft
D 199 ft
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 3
Page 3 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
6. Which choice illustrates a dilation of PQR with a scale factor of 12 ?
y
x0
P R
Q
A
y
x0
Q′P′ R′
B
y
x0P′ R′
Q′
C
y
x0P′
Q′
R′
D y
x0
P′
Q′R′
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 3
Page 4 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
7. A triangle has the following vertices: ( ) ( ) ( )— —1, 1 , 6, 2 , and 3, 5 . If the triangle undergoes a dilation with a scale factor of 3, what will be the vertices of the image?
A ( ) ( ) ( )— —3, 3 , 18, 6 , 9,15
B ( ) ( ) ( )3, 3 , 18, 6 , 9, 15
C ( ) ( ) ( )—3, 3 , 18, 6 , 9, 15
D ( ) ( ) ( )—3, 3 , 18, 6 , 9, 15
8. Figure S is the result of a dilation of Figure T.
+1
+2
+3
+4
+5
+6
0 +1 +2 +3 +4 +5 +6
y
x
Figure T
+1
+2
+3
+4
+5
+6
0 +1 +2 +3 +4 +5 +6
y
x
Figure S
What is the scale factor of the dilation?
A 13
B 12
C 2
D 3
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 3
Page 5 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
9. Rhombus PQRT is shown below.
x
y
0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
— 1
— 2
— 3
— 4
— 5
— 6
+ 1 + 2 + 3 + 4 + 5 + 6— 1— 2— 3— 4— 5— 6
P
Q
R
T
P Q R T′ ′ ′ ′ is the image produced by dilating PQRT by a scale factor of 4. What is the length of the diagonal ?P R′ ′
A 2 units
B 8 units
C 12 units
D 32 units
End of Goal 3 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
Gr08 Math Goal 3 Sample Items Key Report
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 3.01 Represent problem situations with geometric models. Thinking Skill: Organizing Correct Answer: D 2 Objective: 3.02 Apply geometric properties and relationships, including the Pythagorean theorem, to
solve problems. Thinking Skill: Applying Correct Answer: C 3 Objective: 3.02 Apply geometric properties and relationships, including the Pythagorean theorem, to
solve problems. Thinking Skill: Applying Correct Answer: B 4 Objective: 3.02 Apply geometric properties and relationships, including the Pythagorean theorem, to
solve problems. Thinking Skill: Applying Correct Answer: D 5 Objective: 3.02 Apply geometric properties and relationships, including the Pythagorean theorem, to
solve problems. Thinking Skill: Applying Correct Answer: C 6 Objective: 3.03 Identify, predict, and describe dilations in the coordinate plane. Thinking Skill: Analyzing Correct Answer: A 7 Objective: 3.03 Identify, predict, and describe dilations in the coordinate plane. Thinking Skill: Applying Correct Answer: A 8 Objective: 3.03 Identify, predict, and describe dilations in the coordinate plane. Thinking Skill: Applying Correct Answer: A 9 Objective: 3.03 Identify, predict, and describe dilations in the coordinate plane. Thinking Skill: Analyzing Correct Answer: D
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 1 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
1. Charlie collected data on the cost of his long-distance phone calls. He displayed his data in a scatterplot.
x0
3.00
3.50
4.00
4.50
2 4 6 8 10 12 14
y
Length of Call (min)
2.50
2.00
1.50W
X Y
Long-Distance Charges
Z
Which point shows the least expensive cost per minute for a long-distance call?
A W
B X
C Y
D Z
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 2 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
2. Felipe is collecting data comparing air conditioning costs to the daily outdoor temperature during the summer of 2004. When Felipe draws his scatterplot, which variable should be used as the dependent variable?
A date
B indoor temperature
C outdoor temperature
D air conditioning costs
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 3 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
3. The scatterplot below shows the number of people at the swimming pool every half hour from 1:00 p.m. until 5:30 p.m.
Swimmers at the Pool
15
10
5
1:30 2:30 3:30 4:30 5:30Hour
From this scatterplot, what conclusion can be made about the number of people at the pool from 1:00 p.m. to 5:30 p.m.?
A The number of people at the pool steadily decreases and shows a negative correlation with time.
B The number of people at the pool steadily decreases and shows a positive correlation with time.
C The number of people at the pool steadily increases and shows a negative correlation with time.
D The number of people at the pool steadily increases and shows a positive correlation with time.
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 4 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
4. The scatterplot shows the average price of a major-league baseball ticket from 1991 to 2000.
19900
5
Year
Baseball Ticket Prices
10
15
20
1992 1994 1996 1998 2000 2002
What correlation, if any, exists in the data?
A positive
B negative
C constant
D none
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 5 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
5. Which relationship is suggested by the scatterplot below?
Amount of TimeSpent Studying (min)
10 20 30 40 50 60 70 80 900
102030405060708090
100
A The amount of time spent studying does not affect a test score.
B the longer the amount of time spent studying, the higher the test score
C the longer the amount of time spent studying, the lower the test score
D the shorter the amount of time spent studying, the higher the test score
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 6 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
6. Jeremy collected data on the elevation and highest temperature of eight cities. He organized his data in a table.
Place Elevation (ft) Highest Temp (°F)
City A
City B
City C
City D
City E
City F
City G
City H
367—178—722
622
676
26
72
49
136
134
129
128
120
122
108
59
What relationship between elevation and highest temperature does Jeremy’s data suggest?
A the higher the elevation, the higher the record temperature
B the lower the elevation, the higher the record temperature
C the lower the elevation, the lower the record temperature
D There is no relationship between elevation and record temperature.
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 7 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
7. Jessica kept track of gas prices for 9 months.
Month
Gas Prices
1.20
1.40
1.60
1.80
2.00
2.20
1 2 3 4 5 6 7 8 9 10 11 120
According to the line of best fit shown, what will be the predicted price per gallon of gasoline in month 13?
A $1.88
B $1.80
C $1.72
D $1.40
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 8 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
8. The scatterplot shows the number of absences in a week for classes of different sizes. Trevor concluded that there is a positive correlation between class size and the number of absences.
15 20 25 30 35 40
2
4
6
8
10
Class Size
School Absences12
Which statement best describes why Trevor’s conclusion was incorrect?
A The largest class does not have the most absences.
B The smallest class does not have the least number of absences.
C The data show no relationship between class size and number of absences.
D The data show a negative relationship between class size and number of absences.
9. Cheryl surveyed 10 of her classmates and asked them what their favorite type of television show was. Of the 10 students surveyed, 7 indicated that drama was their favorite. Cheryl concluded that dramas are the favorite type of television show for 70% of the 2,000 students at her school. Which choice describes the flaw in Cheryl’s study?
A She asked only students at her school.
B She asked only about television shows.
C She asked only television watchers.
D She asked only 10 people.
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 4
Page 9 Published April 2008. May reproduce for instructional and
educational purposes only; not for personal or financial gain.
10. Jodie wants to conduct a survey to find the most popular school lunch among students in her school. Which is the best sample?
A a random group of students eating lunch together
B all the teachers at lunch
C the first twenty-five students that enter the cafeteria
D every other student who walks into the cafeteria for every period
End of Goal 4 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
Gr08 Math Goal 4 Sample Items Key Report
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 4.01 Collect, organize, analyze, and display data (including scatterplots) to solve problems. Thinking Skill: Analyzing Correct Answer: C 2 Objective: 4.01 Collect, organize, analyze, and display data (including scatterplots) to solve problems. Thinking Skill: Organizing Correct Answer: D 3 Objective: 4.01 Collect, organize, analyze, and display data (including scatterplots) to solve problems. Thinking Skill: Organizing Correct Answer: A 4 Objective: 4.01 Collect, organize, analyze, and display data (including scatterplots) to solve problems. Thinking Skill: Organizing Correct Answer: A 5 Objective: 4.02 Approximate a line of best fit for a given scatterplot; explain the meaning of the line as
it relates to the problem and make predictions. Thinking Skill: Analyzing Correct Answer: B 6 Objective: 4.02 Approximate a line of best fit for a given scatterplot; explain the meaning of the line as
it relates to the problem and make predictions. Thinking Skill: Analyzing Correct Answer: D 7 Objective: 4.02 Approximate a line of best fit for a given scatterplot; explain the meaning of the line as
it relates to the problem and make predictions. Thinking Skill: Applying Correct Answer: A 8
Objective: 4.03
Identify misuses of statistical and numerical data. Thinking Skill: Evaluating Correct Answer: C 9 Objective: 4.03 Identify misuses of statistical and numerical data. Thinking Skill: Analyzing Correct Answer: D 10 Objective: 4.03 Identify misuses of statistical and numerical data. Thinking Skill: Applying Correct Answer: D
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1. Which set of ordered pairs represents a linear relationship?
A ( ) ( ) ( ) ( ){ }— — —0, 1 , 0, 1 , 1, 1 , 1, 2
B ( ) ( ) ( ) ( ){ }2, 2 , 3, 3 , 4, 3 , 5, 3
C ( ) ( ) ( ) ( ){ }— — — —1, 4 , 1, 0 , 0, 1 , 1, 4
D ( ) ( ) ( ) ( ){ }2, 3 , 3, 4 , 4, 5 , 5, 6
2. Which linear function has a graph that includes all of the points in the table below?
x—3—2
0
1
y
4
3
1
0
A —2 2y x= −
B — 1y x= +
C 1y x= −
D 2 1y x= +
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 2 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
3. Which is the graph of x = y?
A
3
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
—1—2—4—5 + 1 +2 + +4 +5 +6— 3—6 0
y
x
B
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
—1—2—4—5 + 1 +2 +3 + 4 +5 +6— 3—6 0
y
x
C
0
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
—1—2—4—5 +1 +2 +3 + 4 +5 +6— 3—6
y
x
D y
x
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
—1—2—4—5 +1 + 2 +3 +4 +5 +6— 3—6 0
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 3 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
4. In the equation y = 3x, x represents yards and y represents feet. Which is the graph of this equation?
A
Yards
y
1 3 42 5 7 86 9
123456789
10
x0 10
B
Yards
y
1 3 42 5 7 86 9
123456789
10
x0 10
C
Yards
y
1 3 42 5 7 86 9
123456789
10
x0 10
D
Yards
y
1 3 42 5 7 86 9
123456789
10
x0 10
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 4 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
5. Which equation describes the line graphed below?
x
y
+6
+5
+4
+3
+2
+1
—1
2
—3
—4
—5
—6
—1—2—4—5 +1 +2 + 3 +4 + 5 +6— 3—6
—
0
A 0x y- =
B 1—x y- =
C 2 1—x y- =
D 2 3—x y+ =
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 5 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
6. ( )
( )
The price of a large pizza is given bythe formula 1.5 7.50, where
is the price of the pizza and is the number of toppings. What doesthe slope represent?
P t tP t t
= +
A number of toppings
B cost per slice
C cost of each topping
D cost of the pizza with no toppings
7. The cost to rent a truck is $60 per day
plus an additional $0.45 for each mile (m) driven. To rent a handcart, there is an additional cost of $5 per day. Jonathan is going to rent a truck and handcart for 2 days. Which equation shows the total amount (R) Jonathan will pay if he drives m miles?
A $130 $0.45= +R m
B $65 $0.45= +R m
C = +$120 $0.45R m
D = +$0.45 $130R m
8.
— —
What are the coordinates of the intercept for the line that goes
through points ( 3, 2) and(3, 6)?
x-
A ( )—2, 0
B ( )— 32, 0
C ( )34, 0
D ( )2, 0
9. Which line has the greatest slope?
A 4 6x y+ =
B 4 6x y− =
C 3 8 1x y− =
D 2 10 3x y− =
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 6 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
10. Which is the graph of the equation y = x — 2?
A
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
0—1—2—4—5 +1 +2 + 3 +4 +5 +6— 3—6x
y
B
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
0—1—2—4—5 +1 +2 + 3 +4 +5 +6— 3—6
y
x
C
+6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
0—1—2—4—5 + 1 +2 +3 +4 +5 +6— 3—6
y
x
D +6
+5
+4
+3
+2
+1
—1
—2
—3
—4
—5
—6
0—1—2—4—5 +1 +2 + 3 +4 +5 +6— 3—6x
y
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 7 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
11.
—
Which is an equation of the line that passes through the points ( 2, 4) and (5, 3)?
A —7 4y x= +
B 7 3y x= +
C 1 267 7
y x= -
D 1 267 7
y x-
= +
12. 23
A line has a slope of and a intercept
of 4. Which of the following is anequation of the line?
—
y-
A 2 3 12x y- =
B 2 3 4—x y- =
C 3 2 4—x y- =
D 3 2 12x y- =
13. Which is an equation of the line that
has a slope of 23
- and passes through
the origin?
A 2 3 0x y+ =
B 3 2 0x y+ =
C 2 3 0x y- =
D 3 2 0x y- =
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 8 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
14. Which equation describes the data in the table below?
5
7
6
15
—
—
4
11
—
—
2
7
—
—
1
1—
x (% reduction [or increase] in dietary fat)
y (weight loss [or gain] in pounds)
A 2 27—x y+ =
B 3x y- =
C 21—x y+ =
D 2 3x y- =
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 9 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
15. The perimeter of a rectangular swimming pool is 42 m. The length is 5 meters more than the width. What is the length of the swimming pool?
A 8 m
B 10.5 m
C 13 m
D 16 m
16. A spring stretches linearly as weight
is added. The table shows data collected for a certain spring.
Weight in lb (x)
100
500
800
900
1,200
Stretch in cm (y)
0.5
2.5
4.0
4.5
6.0
What is the slope of the line that fits these data?
A 1200
B 1100
C 150
D 12
17. Which value satisfies the inequality —2 14 5 2 8?x x x+ − < +
A —3
B 2
C 6
D 7
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 10 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
18. The graph of 2 3 is shown.y x£ -
0x
y
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
— 1
— 2
— 3
+ 1 + 2 + 3 + 4 + 5— 1— 2— 3— 4— 5— 6 + 6
— 4
—5
— 6
Which set contains only points that satisfy the inequality?
A ( ) ( ) ( ) ( ){ }— — — —3, 3 , 4, 11 , 1, 8 , 5, 0
B ( ) ( ) ( ) ( ){ }— — — — —5, 7 , 3, 10 , 5, 7 , 1, 4
C ( ) ( ) ( ) ( ){ }— — — — —1, 10 , 5, 8 , 4, 13 , 3, 2
D ( ) ( ) ( ) ( ){ }— — — —4, 12 , 1, 5 , 3, 4 , 5, 6
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 11 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
19. Sally’s mother said Sally can spend, at most, $25.00 on books and magazines. Books cost $3.00 each, and magazines cost $1.60 each. Which inequality represents the number of books, b, and magazines, m, Sally may purchase?
A 3 1.6 25b m+ ≥
B 3 1.6 25b m+ ≤
C 4.6 25bm ≥
D 4.6 25bm ≤
20. Solve for f:
= −4 2e g f
A 12
2f g e= +
B 2f g e= −
C 12
2f g e= −
D 2f g e= +
21. 2What are the solutions for 4 0?x - =
A —{0, 4}
B —{ 4, 2}
C —{ 2, 2}
D {0, 2}
22. A cube has a volume of 729 3cm . What is the length of each edge of the cube?
A 9 cm
B 11 cm
C 121.5 cm
D 243 cm
23. If −= 56—7
ws and 6,s = what is the
value of w?
A —57
B —9
C 7
D 14
North Carolina Testing Program EOG Grade 8 Math Sample Items Goal 5
Page 12 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
24. 5 2Solve: 15 5
x x+ =
A —1x =
B 15
x−
=
C = 15
x
D 1x =
End of Goal 5 Sample Items
In compliance with federal law, including the provisions of Title IX of the Education Amendments of 1972, the Department of Public Instruction does not discriminate on the basis of race, sex, religion, color, national or ethnic origin, age, disability, or military service in its policies, programs, activities, admissions or employment.
Gr08 Math Goal 5 Sample Items Key Report
Page 1 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
1 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Organizing Correct Answer: D 2 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Applying Correct Answer: B 3 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Applying Correct Answer: C 4 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Applying Correct Answer: D 5 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Analyzing Correct Answer: C
Gr08 Math Goal 5 Sample Items Key Report
Page 2 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
6 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Analyzing Correct Answer: C 7 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Analyzing Correct Answer: A 8 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Analyzing Correct Answer: B 9 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Analyzing Correct Answer: C 10 Objective: 5.01 Develop an understanding of function. A) Translate among verbal, tabular, graphic,
and algebraic representations of functions. B) Identify relations and functions as linear or nonlinear. C) Find, identify, and interpret the slope (rate of change) and intercepts of a linear function. D) Interpret and compare properties of linear functions from tables, graphs, or equations.
Thinking Skill: Applying Correct Answer: A
Gr08 Math Goal 5 Sample Items Key Report
Page 3 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
11 Objective: 5.02 Write an equation of a linear relationship given: two points, the slope and one point on
the line, or the slope and y-intercept. Thinking Skill: Applying Correct Answer: D 12 Objective: 5.02 Write an equation of a linear relationship given: two points, the slope and one point on
the line, or the slope and y-intercept. Thinking Skill: Integrating Correct Answer: A 13 Objective: 5.02 Write an equation of a linear relationship given: two points, the slope and one point on
the line, or the slope and y-intercept. Thinking Skill: Applying Correct Answer: A 14 Objective: 5.02 Write an equation of a linear relationship given: two points, the slope and one point on
the line, or the slope and y-intercept. Thinking Skill: Applying Correct Answer: D 15 Objective: 5.03 Solve problems using linear equations and inequalities; justify symbolically and
graphically. Thinking Skill: Applying Correct Answer: C 16 Objective: 5.03 Solve problems using linear equations and inequalities; justify symbolically and
graphically. Thinking Skill: Applying Correct Answer: A 17 Objective: 5.03 Solve problems using linear equations and inequalities; justify symbolically and
graphically. Thinking Skill: Applying Correct Answer: D 18 Objective: 5.03 Solve problems using linear equations and inequalities; justify symbolically and
graphically. Thinking Skill: Analyzing Correct Answer: A
Gr08 Math Goal 5 Sample Items Key Report
Page 4 Published April 2008. May reproduce for instructional and educational purposes only; not for personal or financial gain.
19 Objective: 5.03 Solve problems using linear equations and inequalities; justify symbolically and
graphically. Thinking Skill: Applying Correct Answer: B 20 Objective: 5.04 Solve equations using the inverse relationships of addition and subtraction,
multiplication and division, squares and square roots, and cubes and cube roots. Thinking Skill: Applying Correct Answer: C 21 Objective: 5.04 Solve equations using the inverse relationships of addition and subtraction,
multiplication and division, squares and square roots, and cubes and cube roots. Thinking Skill: Applying Correct Answer: C 22 Objective: 5.04 Solve equations using the inverse relationships of addition and subtraction,
multiplication and division, squares and square roots, and cubes and cube roots. Thinking Skill: Applying Correct Answer: A 23 Objective: 5.04 Solve equations using the inverse relationships of addition and subtraction,
multiplication and division, squares and square roots, and cubes and cube roots. Thinking Skill: Applying Correct Answer: D 24 Objective: 5.04 Solve equations using the inverse relationships of addition and subtraction,
multiplication and division, squares and square roots, and cubes and cube roots. Thinking Skill: Applying Correct Answer: A
