REVIEW UNIT PROBLEM SETS PROBLEM SET #1 Slopes ...
Transcript of REVIEW UNIT PROBLEM SETS PROBLEM SET #1 Slopes ...
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REVIEW UNIT PROBLEM SETS
PROBLEM SET #1 – Slopes ***Calculators Not Allowed***
Calculate the slope of the line containing the following points:
1. (2,8) 𝑎𝑛𝑑 (−4,6) 2. (−4, −7) 𝑎𝑛𝑑 (3,0) 3. (−3, −6) 𝑎𝑛𝑑 (−1, −6) 4. (4, −2) 𝑎𝑛𝑑 (4,5)
5. (4
5, 6) 𝑎𝑛𝑑 (
3
5, 4)
6. (3
2, −4) 𝑎𝑛𝑑 (2,0)
7. (11
14,
3
7) 𝑎𝑛𝑑 (
9
14,
5
7)
8. (8
9,
2
3) 𝑎𝑛𝑑 (
5
6,
2
5 )
9. (−4, −3) 𝑎𝑛𝑑 (0, −11)
10. (−3
7,
3
8) 𝑎𝑛𝑑 (−
1
6,
5
6)
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PROBLEM SET #2 – Equations of Lines ***Calculators Not Allowed***
For each of the following questions, write the equation of the line given the specific information. 1. Passes through (2,3) and 𝑚 = 2
2. Passes through (−2,4) and 𝑚 =1
2
3. Passes through (−4, −5) 𝑎𝑛𝑑 (2,7) 4. Passes through (3, −5) 𝑎𝑛𝑑 (−3,5)
5. Passes through (-1, 2) and 𝑚 = 0
6. Passes through (-1, 2) and the slope is undefined.
7. Passes through (-2, 2) and is parallel to 2𝑦 =4𝑥 − 12 8. Passes through (-3, 2) and is perpendicular
to 15𝑦 = 10𝑥 + 2
9. 𝑚 =3
5 𝑎𝑛𝑑 𝑏 = 0
10. 𝑚 = 0 𝑎𝑛𝑑 𝑏 = −1
7
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Problem Set #3 – Functions & Graphing Functions ***Calculators Not Allowed***
1. a) True/False 3𝑥2 + 5𝑦 = 7 − 2𝑥 is a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________
2. a) True/False 2𝑥2 + 3𝑦2 = 11 is a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________
3. a) True/False The following table represents a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________
4. a) True/False The following table represents a function. b) Why or why not? ______________________________________________________________ ___________________________________________________________________________
5. Evaluate 𝑔(2) if 𝑔(𝑥) = 3𝑥2 − 5𝑥 + 5
6. Evaluate 𝑔(−3) if 𝑔(𝑥) = −𝑥2 − 2𝑥 + 15
7. Evaluate 𝑓(𝑥 − 2) if 𝑓(𝑥) = −2𝑥2 − 3𝑥 + 11
𝑥 −2 −1 0 1 2 5
𝑦 5 3 6 3 2 −4
𝑥 −2 2 0 −2 2 5
𝑦 4 3 6 3 2 −3
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8. Evaluate 𝑓(5 − 𝑥) if 𝑓(𝑥) =4𝑥−3
2−𝑥
9. Write the new equation of the function 𝑦 = √𝑥 with the following transformations: reflection over x-axis, vertical compression of 1/2, right 3 units, and up 4 units
10. Write the new equation of the function 𝑦 =1
𝑥 with the following transformations:
horizontal compression of 3, right 3 units, and down 2 units
11. Write the new equation of the function 𝑦 = 𝑥2 with the following transformations: reflection over x-axis, vertical stretch of 2, left 2 units, and down 3 units
12. Write the new equation of the function 𝑦 = ln 𝑥 with the following transformations: vertical stretch of 3, right 5 units, and up 2 units
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Problem Set #4 – Piecewise Functions ***Calculators Not Allowed***
1. Graph the following piecewise function:
𝑓(𝑥) = {𝑥 − 1 − 5 ≤ 𝑥 < −1−2 − 1 ≤ 𝑥 < 2−𝑥 + 3 2 < 𝑥 ≤ 6
2. Graph the following piecewise function:
𝑓(𝑥) = {2𝑥 + 1 𝑥 < 1
−𝑥2 + 5 𝑥 ≥ 1
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3. Graph the following piecewise function:
𝑓(𝑥) = {−2𝑥 − 6 𝑥 < −4
𝑥 + 4 − 4 ≤ 𝑥 < 2 𝑥2 − 3 𝑥 ≥ 2
4. Graph the following piecewise function:
𝑓(𝑥) = {|𝑥 + 3| − 1 − 6 ≤ 𝑥 < 1
−𝑥 − 2 𝑥 > 1
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5. Given:
𝑓(𝑥) = { 𝑥2 − 5 𝑥 < −5
11 − 5 ≤ 𝑥 < 1−3𝑥2 + 10 𝑥 ≥ 1
a) Find 𝑓(−7)
b) Find 𝑓(−5)
c) Find 𝑓(0)
d) Find 𝑓(1)
e) Find 𝑓(3)
6. Given:
𝑓(𝑥) = { |𝑥 − 5| + 3 𝑥 < −2
2𝑥3 − 4 𝑥 ≥ −2
a) Find 𝑓(−5)
b) Find 𝑓(−2)
c) Find 𝑓(0)
d) Find 𝑓(1)
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Problem Set #5 – Function Composition ***Calculators Not Allowed***
Use the following functions to answer questions 1 – 16.
𝒇(𝒙) = 𝟑𝒙 𝒈(𝒙) = −𝟐𝒙 − 𝟏 𝒉(𝒙) = |𝒙 − 𝟑| 𝒌(𝒙) = −𝟒𝒙𝟐
1. 𝑔(𝑓(2)) =
2. 𝑓 ∘ 𝑔(3) =
3. ℎ (𝑓(𝑔(0))) =
4. 𝑘 ∘ 𝑔 ∘ ℎ(−2) =
5. 𝑔 ∘ 𝑘(𝑥) =
6. 5𝑓(𝑥) − 3𝑔(𝑥) =
7. 𝑔 ∘ 𝑓 ∘ 𝑘(𝑥) =
8. 𝑓(𝑥)
𝑔(𝑥)=
9. 𝑓(𝑘(3)) =
10.ℎ ∘ 𝑓(−7) =
11. 𝑘 (𝑓(ℎ(−5))) =
12. ℎ ∘ 𝑘 ∘ 𝑔(−1) =
13. 𝑘 ∘ 𝑓(𝑥) =
14. −2𝑔(𝑥) + 4𝑓(𝑥) =
15. 𝑔 ∘ 𝑘 ∘ 𝑓(𝑥) =
16. 𝑔(𝑥)
𝑘(𝑥)=
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Problem Set #6 – Function Roots ***Calculators Not Allowed***
Find any real roots, if they exist, for questions 1 – 12.
1. 𝑦 = 𝑥2 − 2𝑥 − 8
2. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 32
3. 𝑟(𝑡) = 𝑡3 − 11𝑡2 + 18𝑡
4. 𝑦 = −3𝑥2 − 10𝑥 + 8
5. 𝑟(𝑡) = 𝑡3 − 5𝑡2 + 12𝑡
6. 𝑟(𝑡) = 𝑡2 − 6𝑡 + 17
7. 𝑦 = 2𝑥2 − 𝑥 − 10
8. 𝑓(𝑥) = −𝑥2 + 4𝑥 + 12
9. 𝑓(𝑥) = 5𝑥2 + 5𝑥 + 12
10. 𝑦 = 3𝑥2 − 8𝑥 − 2
11.𝑞(𝑧) = 5𝑧3 + 2𝑧2 − 7𝑧
12. 𝑦 = 3𝑥3 + 6𝑥2 − 𝑥
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Problem Set #7 – Domain & Range ***Calculators Not Allowed***
1.
Domain: ____________________ Range: ____________________ 2.
Domain: ____________________ Range: ____________________ 3.
Domain: ____________________ Range: ____________________
4.
Domain: ____________________ Range: ____________________ 5. Domain: ____________________ Range: ____________________ 6.
Domain: ____________________ Range: ____________________
PMI AP Calculus AB NJCTL.org
7. 𝑦 = −2𝑥 + 12 Domain: ____________________ Range: ____________________
8. 𝑦 = 𝑥2 + 4𝑥 − 32 Domain: ____________________ Range: ____________________
9. 𝑦 = −3𝑥2 + 6𝑥 + 5 Domain: ____________________ Range: ____________________
10. 𝑦 = √𝑥 + 5 − 2 Domain: ____________________ Range: ____________________
11. 𝑦 = −√𝑥 + 7 + 5 Domain: ____________________ Range: ____________________
12. 𝑦 =5𝑥+2
√𝑥+3
Domain only: ____________________
13. 𝑦 = ln(𝑥 − 3) Domain: ____________________ Range: ____________________ 14. 𝑦 = 4 ln(𝑥 + 2) − 1 Domain: ____________________ Range: ____________________
15. 𝑦 = −𝑥3 + 14 Domain: ____________________ Range: ____________________
16. 𝑦 = √𝑥 − 8 3
+ 4 Domain: ____________________ Range: ____________________
17. 𝑦 =2𝑥
𝑥2+2𝑥−8
Domain: ____________________ Range: ____________________
18. 𝑦 =25
2𝑥2+5𝑥−3+ 4
Domain only: ____________________
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Problem Set #8 – Inverses ***Calculators Not Allowed***
State whether the following functions are inverses.
1. 𝑓(𝑥) = (𝑥 − 1)2
𝑔(𝑥) = 1 + 𝑥2
2. 𝑔(𝑧) =3
𝑧+ 5
𝑓(𝑧) =3
𝑧−5
3. 𝑔(𝑥) = √𝑥 − 3 +5
ℎ(𝑥) = (𝑥 − 5)2 + 3
4. 𝑘(𝑡) = 2𝑡3 − 1
𝑚(𝑡) =√𝑡+1
3
2
Find the inverse of each function.
5. ℎ(𝑥) = 4√𝑥3
+ 2
6. 𝑘(𝑡) = −5𝑡 + 11
7. 𝑚(𝑥) = 7𝑥2 − 4
8. 𝑔(𝑧) = (𝑧 − 3)5 + 2
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Problem Set #9 – Trigonometry ***Calculators Not Allowed***
Evaluate each of the following.
1. csc7𝜋
6
2. tan𝜋
3
3. sin7𝜋
4
4. 𝑠𝑒𝑐𝜋
6
5. 𝑐𝑜𝑡𝜋
11. 𝑐𝑠𝑐15𝜋
4
12. 𝑐𝑜𝑡2𝜋
4
13. 𝑠𝑖𝑛 4𝜋
3
14. 𝑐𝑠𝑐4𝜋
3
15. 𝑐𝑜𝑠11𝜋
6
6. csc𝜋
4
7. sin𝜋
2
8. cos5𝜋
3
9. 𝑐𝑠𝑐14𝜋
6
10. 𝑡𝑎𝑛2𝜋
3
16. 𝑐𝑜𝑡4𝜋
3
17. 𝑡𝑎𝑛𝜋
2
18. 𝑐𝑜𝑡5𝜋
4
19. 𝑐𝑠𝑐3𝜋
4
20. 𝑐𝑜𝑠5𝜋
2
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Evaluate:
21. 𝑐𝑜𝑠−1 (−√3
2)
22. sin−1( 0)
27. 3 + 2 cos2 (3𝜋
2)
28. cot−1(−1)
23. 𝑐𝑠𝑐−1 (2√3
3)
24.tan−1(−√3
3)
25.sin−1(1)
29. 𝑠𝑒𝑐−1( √2)
30. 2 − 3 sin2 (𝜋
2)
31. cos−1 (−1
2)
26. 𝑐𝑜𝑠−1(0)
32.csc−1(1)
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Problem Set #10 – Exponents ***Calculators Not Allowed***
Simplify:
1. 15𝑚11𝑘−5
10𝑚4𝑘−12
2. 21𝑒3𝑓14
42𝑒7𝑓−3
3. (3𝑥2 − 5𝑥 + 2)(𝑥2 + 3𝑥 − 1)
4. (2𝑦3 + 3𝑦2 − 4)(𝑦2 + 7𝑦 − 3)
5. ((2𝑎4𝑏2)3
(4𝑎9𝑏−5)2)−3
6. ((3𝑚3𝑛4)4
(6𝑚−8𝑛12)2)−2
7. (−5𝑥3𝑦−6𝑧4)−3
8. (4𝑚−2𝑘4𝑝)−2
9. (2𝑥3𝑦3𝑧)2(15𝑥10𝑦4𝑧0)
10. (−5𝑟5𝑠−2𝑡4)2(3𝑟𝑠5𝑡2)
11. (5𝑎 − 2𝑏)2
12. (𝑐 − 4)3
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Problem Set #11 – Logarithms ***Calculators Allowed***
Solve the following equations:
1. log𝑥 16 = 4
2. log𝑥 125 = 3
3. 33𝑥+2 = 108
4. 24𝑥−3 = 12 5. log(7𝑥 + 3) = log (2𝑥 + 23)
6. log(2𝑥 + 3) = log (12𝑥 − 1)
7. 53𝑥 = 26
8. 42𝑥 = 54
9. log2(𝑟 + 3) + log2(𝑟) = log2 10
10. log4(𝑟 + 5) − log4(𝑟) = log4 10
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REVIEW PROBLEM SET ANSWER KEYS
Problem Set #1 – Slopes
1. 1
3
2. 1
3. 0
4. 𝑢𝑛𝑑𝑒𝑓
5. 10
6. 8
7. −2
8. 24
5
9. −2
10. 7
4
Problem Set #2 – Eqns. of Lines
1. 𝑦 − 3 = 2(𝑥 − 2) 𝑜𝑟 𝑦 = 2𝑥 − 1
2. 𝑦 − 4 =1
2(𝑥 + 2) 𝑜𝑟 𝑦 =
1
2𝑥 + 5
3. 𝑦 + 5 = 2(𝑥 + 4) 𝑜𝑟 𝑦 = 2𝑥 + 3
4. 𝑦 + 5 = −5
3(𝑥 − 3) 𝑜𝑟 𝑜𝑟 𝑦 = −
5
3𝑥
5. 𝑦 = 2
6. 𝑥 = −1
7. 𝑦 = 2𝑥 + 6
8. 𝑦 − 2 = −3
2(𝑥 + 3) 𝑜𝑟 𝑦 = −
3
2𝑥 −
5
2
9. 𝑦 =3
5𝑥
10. 𝑦 = −1
7
Problem Set #3 – Functions/Graphing
1. a) TRUE b) Each x-value corresponds to
only one y-value.
2. a) FALSE b) Does not pass vertical line
test; more than one y-value for each x-value
3. a) TRUE b) Each x-value corresponds to
only one y-value
4. a) FALSE b) Does not pass vertical line
test; more than one y-value for each x-value
5. 7
6. 12
7. −2𝑥2 + 5𝑥 + 9
8. 4𝑥−17
3−𝑥
9. 𝑦 = −1
2√𝑥 − 3 + 4
10. 𝑦 =1
3𝑥−9− 2
11. 𝑦 = −2(𝑥 + 2)2 − 3
12. 𝑦 = 3ln(𝑥 − 5) + 2
Problem Set #4 – Piecewise Functions
1. See graph
2. See graph
3. See graph
4. See graph
5. a) 44 b) 11 c) 11 d) 7 e) -17
6. a) 13 b) -20 c) -4 d) -2
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Problem Set #5 – Function Composition
1. −13
2. −21
3. 6
4. −484
5. 8𝑥2 − 1
6. 21𝑥 + 3
7. 24𝑥2 − 1
8. 3𝑥
−2𝑥−1
9. −108
10. 24
11. −2304
12. 7
13. −36𝑥2
14. 16𝑥 + 2
15. 72𝑥2 − 1
16. 2𝑥+1
4𝑥2
Problem Set #6 – Function Roots
1. 𝑥 = −2 & 𝑥 = 4
2. 𝑥 = −8 & 𝑥 = 4
3. 𝑡 = 0 & 𝑡 = 2 & 𝑡 = 9
4. 𝑥 =2
3 & 𝑥 = −4
5. 𝑡 = 0
6. no real roots
7. 𝑥 = −2 & 𝑥 =5
2
8. 𝑥 = −2 & 𝑥 = 6
9. no real roots
10. 𝑥 =4±√22
3
11. 𝑥 = 0, 𝑥 = −7
5 & 𝑥 = 1
12. 𝑥 =−3±2√3
3 𝑎𝑛𝑑 𝑥 = 0
Problem Set #7 – Domain & Range
1. Domain: (−∞, 1) ∪ [4, ∞)
Range: ℝ
2. Domain: ℝ
Range: (−∞, 3]
3. Domain: [−5,5]
Range: [−2,2]
4. Domain: ℝ
Range: 𝑦 = 3
5. Domain: (−∞, −3] ∪ (−2, ∞)
Range: (−∞, 3]
6. Domain: (−∞, 2] ∪ (3, ∞)
Range: 𝑦 = −2 𝑎𝑛𝑑 (−1, ∞)
7. Domain: ℝ
Range: ℝ
8. Domain: ℝ
Range: [−36, ∞)
9. Domain: ℝ
Range: (−∞, 8]
10. Domain: [−5, ∞)
Range: [−2, ∞)
11. Domain: [−7, ∞)
Range: (−∞, 5]
12. Domain: (−3, ∞)
13. Domain: (3, ∞)
Range: ℝ
14. Domain: (−2, ∞)
Range: ℝ
15. Domain: ℝ
Range: ℝ
16. Domain: ℝ
Range: ℝ
17. Domain: ℝ 𝑥 ≠ 2 𝑥 ≠ −4
Range: (−∞, 0) ∪ (0, ∞)
18. Domain: ℝ 𝑥 ≠1
2 𝑥 ≠ −3
Problem Set #8 – Inverses
1. No
2. Yes
3. Yes
4. No
5. ℎ−1(𝑥) = (𝑥−2
4)3
6. 𝑘−1(𝑡) =11−𝑡
5
7. 𝑚−1(𝑥) = √𝑥+4
7
8. 𝑔−1(𝑧) = √𝑧 − 25
+ 3
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Problem Set #9 – Trigonometry
1. −2
2. √3
3. −√2
2
4. 2√3
3
5. 𝑢𝑛𝑑𝑒𝑓
6. √2
7. 1
8. 1
2
9. 2√3
3
10. −√3
11. −√2
12. 0
13. −√3
2
14. −2√3
3
15. √3
2
16. √3
3
17. 𝑢𝑛𝑑𝑒𝑓
18. 1
19. √2
20. 0
21. 5𝜋
6
22. 0
23. 𝜋
3
24. −𝜋
6
25. 𝜋
2
26. 𝜋
2
27. 3
28. −𝜋
4
29. 𝜋
4
30. −1
31. 2𝜋
3
32. 𝜋
2
Problem Set #10 – Exponents
1. 3𝑚7𝑘7
2
2. 𝑓17
2𝑒4
3. 3𝑥4 + 4𝑥3 − 16𝑥2 + 11𝑥 − 2
4. 2𝑦5 + 17𝑦4 + 15𝑦3 − 13𝑦2 − 28𝑦 + 12
5. 8𝑎18
𝑏48
6. 16𝑛16
81𝑚56
7. −𝑦18
125𝑥9𝑧12
8. 𝑚4
16𝑘8𝑝2
9. 60𝑥16𝑦10𝑧2
10. 75𝑟11𝑠𝑡10
11. 25𝑎2 − 20𝑎𝑏 + 4𝑏2
12. 𝑐3 − 12𝑐2 + 48𝑐 + 64
Problem Set #11 – Logarithms
1. 𝑥 = 2
2. 𝑥 = 5
3. 𝑥 = 0.754 𝑜𝑟 0.753
4. 𝑥 = 1.646
5. 𝑥 = 4
6. 𝑥 = 0.4
7. 𝑥 = 0.861
8. 𝑥 = 2.322 𝑜𝑟 2.321
9. 𝑟 = 2
10. 𝑟 =5
9