sect 3.1 reading graphs how much money will you earn in a lifetime with an associate’s degree?...

Sect 3.1 Reading Graphs

How much money will you earn in a lifetime with an associate’s degree?

What degree must you obtain to earn at least 2 million dollars in a lifetime?

$1.5 to 1.6 million

Bachelor’s or higher.


of

is

100

%

8.134100

9 x

1008.1349 x

billionx 132.12

000,387,5000,000,132,12. Avg

09.252,2$. Avg

Find the amount of 134.8 billion that was given to Pell Grants.


How many months of regular exercise are required to lower the pulse rate as much as possible?

How many months of regular exercise are needed to achieve a pulse rate of 65 beats per min.?

Indicates no visual comparisons!

6 MONTHS

3 MONTHS


All points are labeled (x, y).

x

y

__)(__,A

__)(__,B

__)(__,C

__)(__,D

__)(__,E

__)(__,F

__)(__,G

),( yxP

5

4

4

3

3

3

42

51

2 0

0 3


These points are not in a Quadrant!

Sect 3.2 Graphing Linear Equations

12 xy

Determine if (2, 5) and (-1, -1) are solutions to

125,2 xy

1225 145

55

121,1 xy

1121 121

11

TRUEYes (2, 5) is a solution.

TRUEYes (-1, -1) is a solution.

(2, 5)

(-1, -1)


23 xyGraph

x (x, y)23 xy

When the equation is solved for y, y is the dependent variable and x is the independent variable. This means we can pick values for x and substitute them in to find the y value.

Start with x = 0. Easiest to multiply by.

2

203

y

y0 2,0

1

213

y

y1 1,1

4

223

y

y2 4,2

Use a straight edge to connect the points to form a straight line.


32 xyGraph

x (x, y)32 xy

Start with x = 0. Easiest to multiply by.

3

302

y

y0 3,0

1

312

y

y1 1,1

1

322

y

y2 1,2



13

2 xyGraph

x (x, y)13

2 xy

Start with x = 0, Count by 3’s due to the denominator in the fraction.

1

1032

y

y0 1,0

1

12

1332

y

y

y3 1,3

3

14

1632

y

y

y6 3,6

Notice a pattern in the points?Y-coordinates and X-coordinates.

+2

+2+3

+3

The fraction contains the changes in the y-coord. and x-coord.!

Sect 3.2 Graphing Linear Equations3

2

1 xyGraph

x (x, y)32

1 xy

Start with x = 0, Count by 2’s due to the denominator in the fraction.

3

3021

y

y0 3,0

2

31

3221

y

y

y2 2,2

1

32

3421

y

y

y4 1,4


-1

-1+2

+2

Sect 3.2 Graphing Linear Equations552 yxGraph

x (x, y)15

2 xy

Solve the equation for y!

yx 552 + 5y + 5y

yx 15

2 5 5 5

1

1052

y

y0 1,0

3

12

1552

y

y

y5 3,5

1

12

1552

y

y

y5 1,5

+ 5 + 5

Sect 3.3 Graphing Linear Equations in Standard Form

552 yxGraph

x-intercept ( ____, 0 )

y-intercept ( 0, ____ )

B

AmCByAx

;

5

2

5

2

m

5052 x52 x

2

5x

2

5

5502 y55 y

1y

1


Graph



1242 yx

2

1

4

2

m

12042 x122 x

6x

6

12402 y124 y

3y

3


Graph



1223 yx

2

3

2

3

m

12023 x123 x

4x

4

12203 y122 y

6y

6


Graph



054 yx

0054 x04 x

0x

0

0504 y05 y

0y

0

5

4

5

4

m


Graph



yx 884

Rewrite in standard form: Ax + By = C

884 yxShort Cut: Cover-up technique.

To find the x-intercept, cover-up the y term.

Solve for x.

2x

2

To find the y-intercept, cover-up the x term.

Solve for y.

1y

1

2

1

8

4

m


Graph



1132 x

Notice there is no y variable in the equation…the line can’t cross the y-axis. No y-intercept.

82 x4x

4


Graph



63 y

Notice there is no x variable in the equation…the line can’t cross the x-axis. No x-intercept.

2y

2

Sect 3.3 Graphing Linear EquationsGraph



1135 yx

5

11

3

11 YUCK!

Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side.

5 3 11x y

Start with 11 and keep subtracting by 5 until you have a number divisible by 3.

11 5 6

3 6, 1y when x 5

3

Am

B

5 5x x 3 11 5y x

2y

We only subtracted by one 5, so x = 1.

1,2

Sect 3.3 Graphing Linear EquationsGraph



1957 yx

7

19

5

19

YUCK!

Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side.

7 5 19x y

Start with 19 and keep subtracting by 5 until you have a number divisible by 5.19 7 12

12 7 5

5 5, 2y when x

7 7

5 5

Am

B

7 7x x 5 19 7y x

1y

We subtracted by two 7’s, so x = 2.

2, 1

Sect 3.4 Rates

A Rate is a ratio that indicates how two quantities change with respect to each other.

Unit rate is when the second quantity is one.

On Jan. 3rd, Joe rented a car with a full tank of gas and 9312 miles on the odometer. On Jan. 7th, he returned the car with 9630 miles on the odometer. If Joe had to pay $108 for the total bill which included 12 gallons of gas, find the following rates. Convert to unit rates.

1. Gas consumption in miles per gallon.

2. Average cost of the rental in dollars per day.

3. Travel rate in miles per day.

miles31893129630

gallons

miles

12

318Divide for unit rate.

gallons

miles5.26

days5

108$

Jan. 3rd, 4th, 5th, 6th, and 7th = 5 daysday

6.21$

days

miles

5

318

day

miles6.63

Sect 3.5 Slope

The Slope is a ratio that indicates how the change in the y-coordinates change with the respect to the change in the x-coordinates.

The slope of a line contains two points ( x1, y1 ) and ( x2, y2 ) is given by

12

12

xx

yy

run

rise

xinchange

yinchangem

11, yx

22 , yxChange in y

Change in x


Graph ( -4, 3 ) and ( 2, -6 ) and find the slope.

2

3

6

9

m

Change in y = -9

Change in x = 6

2

3

6

9

42

3612

12

m

m

xx

yy

run

risem

Sect 3.5 Graphing Linear Equations1m2m3m

3

2m

2

1m

4

1m

Positive slopes always have lines that go uphill.

Slopes > 1 are steep.

0 < Slopes < 1 begin to flatten out.

Sect 3.5 Graphing Linear Equations1m 2m 3m

3

2m

2

1m

4

1m

Negative slopes always have lines that go downhill.

Slopes < -1 are steep.

-1 < Slopes < 0 begin to flatten out.


Graph ( -4, 3 ) and ( 2, 3 ) and find the slope.

Horizontal Line

06

0

42

33

m

m

undefinedm

m

0

755

16

Graph ( 5, -1 ) and ( 5, 6 ) and find the slope.

Vertical Line

Sect 3.5 Slope

The Grade is a slope that is measured as a percent.

fthorizontal

ftvertical

100

7%7

Drop 7 feet

for every 100 feet traveled horizontally.

Sect 3.6 Graphing Linear Equations in Slope Intercept Form

Graph

43

2 xy

( 0, - 4 )Starting point

Directions 2 up & 3 right

Or opposite 2 down & 3 left


Graph

62

5 xy

32 yx

( 0, 6 )Starting point

Directions 5 down & 2 right

+ y = + y

yx 3232 xy

( 0, -3 )Starting point

Directions 2 up & 1 right

Opposite2 down & 1 left

-3 -3

1


Graph

1035 yx

1145 yx

Solving for y would not be a good decision because it will generate a fraction for a y-intercept.

x-intercept and slope.

0,23

5

3

5

B

Am

x-intercept and y-intercept are bad…fractions.

6511 156

451 x = 3 for the three 5’s we subtracted.

4

5

4

5

B

Am

4 11 5y x

4 4y 1; 3,1y

Sect 3.6 Graphing lines

Parallel Lines – Lines are parallel when they have the same slopes or the lines are vertical.

21 mm

Perpendicular Lines – Lines are perpendicular when their slopes are opposite reciprocals of each other.

121 mm

I prefer to say, “the perpendicular slopes are opposite reciprocals.”

Sect 3.7 Graphing Linear Equations in Point Slope Form

24

1

x

ym

Consider a line with slope 2 passing through the point ( 4, 1 ).

( 4, 1 )

( x, y )

Consider removing the “m” and re-writing so there is no fraction.

12

4

y

x

The equation is called point-slope form of a linear equation

with the slope m and the point .

11 xxmyy

11, yx

Notice the sign change!

14 2 4

4

yx x

x

1 2 4y x

Sect 3.7 Graphing Linear Equations in Point Slope Form

Graph

14

35 xy

43

12 xy

42

35 xy

245 xy

Start @ (1, 5) m = -34

Start @ (-4, 2) m = 13

Start @ (-4, -5) m = 32

Start @ (2, -5) m = 41


Write the equation of a line in point-slope form with the slope of and the point ( -14, 12). Convert to slope-intercept form. 7

5

Write the equation of a line in point-slope form with the slope of and the point ( 4, -6). Convert to slope-intercept form. 5

8

11 xxmyy

147

512 xy 10

7

512 xy

147

5

7

512 xy

2 + 12 = + 12

227

5 xy

45

86 xy

5

32

5

86 xy

45

8

5

86 xy

– 6 = – 65

30

5

32

5

8 xy

5

2

5

8 xy

Sect 3.7 Graphing lines…ANOTHER APPROACH

Write the equation of a line in point-slope form with the slope of and the point ( 4, -6). Convert to slope-intercept form. 5

8

8A

NO FRACTION WORK…Remember Standard Form.B

AmCByAx

;

5

8

B

Am

5BCyx 58

Substitute the point ( 4, -6 ) for x and y to solve for C. C 6548

C 3032

C2

Cyx 58

258 yx – 8x = – 8x

285 xy 5 5 5 5

2

5

8 xy

Solve for y.

Short cut.

5

2

B

Cb


Write the equation of a line that is parallel to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form.

Write the equation of a line that is perpendicular to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form.

B

AmCByAx

;

5

3

B

Am

y-intercept (0, b) bmxy

25

3 xy

Same slopes.

Opposite and Reciprocal slopes.

5

3

B

Am

Perpendicular slope.

3

5

A

Bm

y-intercept (0, b)

23

5 xy


Write the equation of a line that is parallel to 7x – 4y = 9 and contains the point ( 5, -3). Use slope-intercept form.

Write the equation of a line that is perpendicular to 8x + 3y = 13 and contains the point ( -7, 4). Use slope-intercept form.

B

AmCByAx

;

Same slopes.Must be the same!.

Cyx 47

C 3457

C1235 47

4747 yx B

Cb

7 47

4 4y x

3

8

B

Am

Perpendicular slope.

8

3

A

BmThe numbers must flip and change the sign!

1338 yx Cyx 83

C 4873

C 3221 53 8

53

8

53

B

Cb

8

53

8

3 xy

5383 yxNew.

4

7

47

4

4774 xy

4

4

47

4

7 xy

4 4

Sect 9.4 Graphing Linear Inequalities

Graph the line .

Consider changing the equation to an inequality.

Determine if the points are solutions, ( -4, 3 ), (-8, 5 ), and ( 2, -1 ).

32

3 xy

32

3 xy

382

35

3125

95 FALSE

322

31

331

61 TRUE

Sect 9.4 Graphing Linear Inequalities in Slope Intercept Form

Graph

62

5 xy

32 yx

The inequality has y > which tells me to shade all the y coordinates greater than the line…above the line.

32 xy

– 2x = – 2x

-1 -1 -1

32 xy

Solve for y.

Remember to flip the inequality symbol.

Shade all the y coordinates less than the line…below the line.

No equal to line means a dashed line!

Equal to line means a solid line!


Graph the line .

Consider changing the equation to an inequality.

Determine which way to shade.

842 yx

842 yx

Test the Origin (0, 0)

80402 80

TRUE

Shade in the direction of the Origin

Find the x and y intercepts.


Graph

2434 yx

1642 yx



240304 240 TRUE



160402 160 FALSE


Graph

246 y

122 x

4y

6x


Graph

23 y

sect 3.1 reading graphs how much money will you earn in a lifetime with an associate’s degree?...

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