X-intercept Roots Zeros Hits ground/water ~= 0 Crosses x-axis (~,0)
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X-intercept Roots ZerosHits ground/water ~=0Crosses x-axis (~,0)
y1= equation y2=0 2nd Trace 5 MOVE Enter Enter Enter
Graphing Method:
Table Method: The x values when y=0 (or y goes from + to -)
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y-intercept Initial Value
Start at
Equation:
Table:
Calculator:
The number without a variabley= 3x2 +5x + 3 or y = 6(2)x
The y value when x = 0
2nd Trace 1(value) 0 Enter
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Exponential Growth (Equation):y=abx or y=a(1 + r)x
y=final amount a=initial amount b=base r=rate as a percentage (4.5%= .045)
If b > 1 then it’s increase; If 0<b<1 then it’s decay
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Decimals of
Depreciate (Decrease) 5%
&Appreciate (Increase) 13.5%
(1- .05) = .95
(1 + .135) = 1.135
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% & % Increase/Decrease
50$ shirt with 7% increase (tax)
50$ shirt with 20% decrease (off)
Final = Beginning ( 1 + r)
50(1.07) $53.50
50(0.8)$40
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Linear Growth:
Slope of y =3x + 2
Y-intercept of y = 3x + 2
y= mx + b m=slope (rate of change, change per x) b=y-intercept (crosses y-axis & initial value)
3 (also rate of change )
2 (also starting value)
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The equation p(x) = .5x + 2 where x is the year since 2000 and p(x) represents the population (in thousands) of Pineboro.What is the slope and what does it represent?
What is the y-intercept and what does it represent?
.5 thousand; increase in population per year
2 population in 2000
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Find rate per mile: $2 per .4 mile
Find rate per mile: $1.20 per .2 mile
Find rate per minute: $3 per 30 seconds
Find rate per foot: $12 per 10 feet
$2/ .4 = $5 per mile
$1.20/.2 = $6 per mile
3/.5 = $6 per minute
$12/10 = 1.2 per foot
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Write equations of the following:Balloon company charges flat-fee of $10 per delivery and $2 per mile
Taxi charges a flat-fee of $3 and $2 per 0.5 mile
Car rental charges $20 a day plus 65 cents a mile
Car rental charges $50 per day
C = 2m + 10
F= 2/.5 m + 3 Or F= 4m + 3
C=0.65m + 20
C = 50
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Regressions & Coefficient of Correlation
Coefficient of correlation:
STAT EDIT Enter x’s into L1 Enter y’s into L2 STAT CALC Choose Linreg or Expregr ENTERWrite equation on paper and then type in y= ~1 Positive ~ -1 Negative Find: 2nd 0 Scroll to DiagnosticON Enter then do regression
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The number of schools in Turnstown can be represented by the following table:
According to the line of best fit, how many schools are predicted for 2014?
Year 1960 1970 1990 2004 2010
Schools 20 40 80 108 120
1) Change 1960 to 0 and the other years (only do on year problems)
2) STAT EDIT Put the numbers in STAT CALC Linregression
3) Write the equation. Y = 2x + 20
4) Type Equation into Y= y = 2x + 205) Look at table for x=54 (2014-1960 = 50) 54 128
0 10 30 44 50
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Exponent Examples
2a2*3a5
(3a5)2
= 2aa 3aaaaa = 6a7
Multiply: Mult. Coefficients, add exponents
= (3aaaaa)(3aaaaa)= 9a10
Power: Write it out then perform rule
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Exponent Examples Continued:
33 2
1
2
1
12
6
aaaaaaaaa
aa
2
3
a
5
2
12
6
a
a
23 aWrite out the variables; reduce the coefficients; cancel letters
Negative exponents: Switch the location and make positive;
5xo = 5 Zero power makes exponent disappear
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2
02
43
zy
yxSimplify
024024
33
zyx
y
zyx
y
02
4
02
4 33
zy
yx
zy
yx
xxxxyy
y
xxxxyy
y 33
48
29
yx
y
28
9
yx
Write it out twice
Move negative exponents
Write out exponents; remove Zero exponent
Combine exponents
Subtract exponents
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432 3612 xxx
432 3612 xxx Simplify
x
21
2
x
Dividing by a monomialDivide each by the bottom
6x3
6x3 6x3 6x3Divide each by 6x3
Simplify
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Greatest Common Factor (GCF):
42x2yz4 and 28x5 y3
•Largest integer that goes into each coefficient
•Smallest of each exponent
14x2 y
Calculator (for coefficient)
Math Num GCD (#1,#2) z
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Least Common Multiple (LCM):
Example: 10x2 yz4 and 15x5 y3
Smallest integer they go into, largest of each exponent
LCM: 30x5y3z4
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Slope of Parallel Lines:
What is the slope of the line parallel to y = 3x + 2?
What is the slope of the line parallel to 3x + 2y = 8
Same slope (5/6 & 5/6)
3
2y = -3x + 8 y = -3/2x + 8/2 so -3/2
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Slope of Perpendicular Lines:
What is the slope of the line perpendicular to y = 3x + 2?
What is the slope of the line perpendicular to 3x + 2y = 8
Negative Reciprocal slope (5/6 & -6/5)
-1/3
2y = -3x + 8 y = -3/2x + 8/2 so 2/3
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Direct Variation: (Set up Proportion then solve)
Y varies directly as x. y is 8 when x is 3. Findx when y is 40
Y = YX X8 = 403 X
8x = 3(40)
x = 120/8 = 15
Write proportion
Substitute in
Cross multiply
Solve by dividing
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The number of dolls made varies directly as the number of hours worked. 100 dolls are made in 25 hours. Find how many dolls are made in 75 hours
dolls = dollshours hours100 = d25 75
25x = 75(100)
x = 7500/25 = 300
Write proportion
Substitute in
Cross multiply
Solve by dividing
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Systems of Equations2x – y = 10 Find x + yx = y + 4
2x – 1y = 101x – 1y = 4
2nd x-1 1 ENTER 2 X 3 [ 2 -1 10 ][ 1 -1 4 ]
2nd MODE 2nd x-1 Scroll to RREF 2nd x-1 1 ENTER
You get [ 1 0 6 ] so x = 6 and y = 2 so x + y = 8 [ 0 1 2 ]
Put in matrix ready mode with x’s and y’s together
Put into matrix
Solve with matrix
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Midpoint of (x1, x2) and (y1, y2)
Midpoint: Mx = My =
Or Graph to find Midpoint
2
)( 21 xx 2
)( 21 yy
Find midpoint of (3,2) and (-1,5)
)5.3,1(2
)52(,
2
)13(
Midpoint is ½ way
(1, 3.5)
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Mary’s house is (3, 1) and is the exactly halfway between Jack’s house (-1,3) and Gary’s house? Find the coordinates of Gary’s house.
Midpoint
Jack’s House
Gary’s House(5, -1)
Down 2Right 4
Right 4
Down 2
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What does An+1=2An + 3 mean?Next number = 2 * (now number) + 3
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The number of butterflies, of a particular day, is related to the previous day’s number of butterflies, Bn by the function Bn+1 = 1.12Bn.
A)Describe the change in butterfliesIncreases by 12% every day(112% - 100% = 12%)B)Find the number of butterflies on the 4th day if there was 50 butterflies on the first day Day 1: 50 Day 2: 50(1.12) = 56 Day 3: 56(1.12) = 62.7 Day 4: 62.7(1.12) = 70.24 butterflies
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Factor trick with Calculator (only for multiple choice)Use if given area or “Which is a factor?
1) 37 STO X ENTER Store 37 as x2) (10x2 – x – 2 ) / (5x – 2) Decimal so not a factor3) (10x2 – x – 2)/(2x + 1) Decimal so not a factor 4) (10x2 – x – 2)/(10x – 3) Decimal so not a factor5) (10x2 – x – 2)/(2x – 1) NOT A DECIMAL so it’s the answer!
Example: Area is 10x2 – x – 2. Which could be a length?A) 5x – 2 B) 2x + 1 C) 10x – 3 D) 2x - 1
1) 37 STO X ENTER Store 37 as x2) (Problem) / (Answer A) If no decimal then it could be a factor
(Problem)/(Answer B)(Problem)/(Answer C)(Problem)/(Answer D)
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Domain
& Independent Variables:
All x values (numbers from left to right)
Domain of {(2,3),(5,6),(1,8)}Domain of y = x2 + 3x
{1,2,5}All reals
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Range
& Dependent Variables:All y values (numbers from down to up)
Range of {(2,3),(5,6),(1,8)}Range of y = x2 + 6x
{3,6,8}y> -9
(-3,-9)
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Error on calculator:
If you have an error that you can’t fix immediately then reset your memory by: 2nd + 7 1 2
If calculator keeps saying 2nd Curve when finding intersection then make sure you have y2 = 0
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Slope of (x1, x2) and (y1, y2)
m=
)(
)(
12
12
xx
yy
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Distance between (x1,x2) &(y1,y2) Distance:
don’t forget to close the parenthesis
)( 22 )yy(+)xx(
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Pythagorean Theorem
Pythagorean Theorem: a2 + b2 = c2
a
b
c
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Area of rectangle and triangle:
Rectangle A=lw
Triangle A = ½ bh
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Area &Circumference of Circle:
Area: A= (3.14) r2
Circumference=2(3.14)r or (3.14)d
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Other Reminders:4 less than a number:
4 is less than a number:
Sum of 3 and x:
Twice the difference of x and 4:
There are twice as many cats as dogs
Quadrants:
x-4
x<4
(x+3)
2(x-4)
c = 2d
II I
III IV