# x-intercept roots zeros hits ground/water ~= 0 crosses x-axis (~,0)

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X-intercept Roots Zeros Hits ground/water ~= 0 Crosses x-axis (~,0). Table Method:. y1= equation y2=0 2nd Trace 5 MOVE Enter Enter Enter . The x values when y=0 (or y goes from + to -). Graphing Method:. y-intercept Initial Value Start at. - PowerPoint PPT PresentationTRANSCRIPT

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 -)

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

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

Decimals of

Depreciate (Decrease) 5%

&Appreciate (Increase) 13.5%

(1- .05) = .95

(1 + .135) = 1.135

% & % 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

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)

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

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

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

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

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

Exponent Examples

2a2*3a5

(3a5)2

= 2aa 3aaaaa = 6a7

Multiply: Mult. Coefficients, add exponents

= (3aaaaa)(3aaaaa)= 9a10

Power: Write it out then perform rule

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

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

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

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

Least Common Multiple (LCM):

Example: 10x2 yz4 and 15x5 y3

Smallest integer they go into, largest of each exponent

LCM: 30x5y3z4

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

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

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

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

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

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)

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

What does An+1=2An + 3 mean?Next number = 2 * (now number) + 3

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

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)

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

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)

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

Slope of (x1, x2) and (y1, y2)

m=

)(

)(

12

12

xx

yy

Distance between (x1,x2) &(y1,y2) Distance:

don’t forget to close the parenthesis

)( 22 )yy(+)xx(

Pythagorean Theorem

Pythagorean Theorem: a2 + b2 = c2

a

b

c

Area of rectangle and triangle:

Rectangle A=lw

Triangle A = ½ bh

Area &Circumference of Circle:

Area: A= (3.14) r2

Circumference=2(3.14)r or (3.14)d

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