12/7/2015math 120 - km1 chapter 3 3.1 cartesian coordinates3.1 3.2 linear functions: graphs and...

04/21/23 Math 120 - KM 1

Chapter 3

3.1 Cartesian Coordinates

• 3.2 Linear Functions: Graphs and Slope

• 3.3 More on Graphing Linear Equations

• 3.4 Finding Equations of Lines; Applications

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The Cartesian PlaneRene Descartes (1596 - 1650)

2.1

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Coordinates of Points

2.1

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Plotting Points

2.1

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Plot: y = 2x - 1

x y

0 -1

1/2 0

1 1

10 ,

0

2

1,

11,

2.1

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Plot: y = -2x

x y

0 0

0 0

1 -2

2 -4

21,

00,

42 ,

2.1

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2.2 Linear Functions: Graphs and

Slope

2.4

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Linear Function: f(x) = mx + b

Remember y is the same as f(x).

f(x) = mx + b or

y = mx + b

(x, y)(x, f(x))

2.4

04/21/23 Math 120 - KM 9


What does b do?

y = x

y = x + 3

y = x - 4

(0,3)

(0,0)

(0,-4)

2.4

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What does m do?

y = x

y = 3xy = -4x

(1,3)

(1,1)

(1,-4)

2.4

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y = mx+b(0, b) is the y-intercept

m is the slope

y = mx + b

(0,b)RUN

1

RISEm

2.4

04/21/23 Math 120 - KM 12

Equation Slope y-intercept

y = 2x - 5

y = -x + 1

y = x

y = 3

x = 2

3x- 2y - 4 = 0

y = mx+bm is the slope

(0, b) is the y-intercept

(0,-5)

(0, 1)

(0,0)

(0,3)

no y-intercept

m = 2

m = -1

m = 1

m = 0

no slope

(0,-2)m = 3/2

2.4

04/21/23 Math 120 - KM 13

SLOPEBASICS

POSITIVE SLOPE The line rises from left to right.

ZERO SLOPE The line is HORIZONTAL (zero rise)

NEGATIVE SLOPEThe line falls from left to right.

UNDEFINED SLOPEThe line is VERTICAL (zero run)

2.4

04/21/23 Math 120 - KM 14

Computing the Slope:(x1,y1) and (x2, y2)

are coordinates of two points

Slope

xy

21

12

12 xx,xxyy

m

xinchange

yinchange

RunRise

xdelta

ydelta

0Run

2.4

04/21/23 Math 120 - KM 15

A Slope Triangle

21

12

12 xx,xxyy

m

P1(x1, y1)

RUNx2 - x1

RISEy2 - y1

P2(x2, y2)

RISE

RUN

2.4

04/21/23 Math 120 - KM 16

Compute the Slope

(-2, -3)

(4, -1)

3

1

6

2

24

31

)(

m

Upwards + Slope

2.4

04/21/23 Math 120 - KM 17

Another Slope

(-4, -3)

(0, 5)

21

2

4

8

40

35

m

Upw

ards

+

Slo

pe

2.4

04/21/23 Math 120 - KM 18

Compute this Slope

(-5, 2)

(4, -1)

3

1

9

3

54

21

)(

m

DOWNWARDS - Slope

2.4

04/21/23 Math 120 - KM 19

Compute the Slope

(-5, -1)

(4, -1)

0

9

0

54

11

)(

m

HORIZONTALO

2.4

04/21/23 Math 120 - KM 20

Compute the Slope

(-4, -5)

(-4, 0)

Undefined

)(m

0

5

44

50

VER

TIC

AL

no s

lope

2.4

04/21/23 Math 120 - KM 21

Practice Problem 1a

3

2

Determine the slope of the line

containing points P1 and P2. ),(P 571 ),(P 912

6

4

71

59

)(

m

RUN = 3

RISE = 2

The line rises from left to right.

2.4

04/21/23 Math 120 - KM 22

Practice Problem 1b



0

7

33

52

)(

m

The slope is undefined.

Vertical Line

2.4

04/21/23 Math 120 - KM 23

Practice Problem 1c

2

1



4

2

04

35

)(

m

RUN = 2

RISE = -1

The line falls from left to right.

2.4

04/21/23 Math 120 - KM 24

Practice Problem 1d

0



6

0

71

55

)(

m

RUN = 6RISE = 0

The line is Horizontal.

2.4

04/21/23 Math 120 - KM 25

2.5 More on Graphing Linear Equations

2.5

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Intercepts

x-intercept ( , 0)

Y – intercept (0 , )

2.5

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Sketch the Graph

x-intercept ( , 0)


(3, 0)

(0, 6)

2.5

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Sketch the Graph

x-intercept ( , 0)


(1, 0)

(1, 3)

2.5

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Sketch the Graph

x-intercept ( , 0)


(3, 0)

(0, -2)

2.5

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Sketch the Graph

x-intercept ( , 0)


(3, 2)(0, 2)

2.5

04/21/23 Math 120 - KM 31

Sketch the Graph

x-intercept ( , 0)


(3, 2)

(0, 0)

2.5

04/21/23 Math 120 - KM 32

Graph a Line using a Point and the

Slope

P1(x1, y1)

P2(x1+ RUN, y1+ RISE)

RUN

RISE

2.5

04/21/23 Math 120 - KM 33

Graph

(0, - 4) is the y-intercept

2/5 is the slopeRun 5 then rise 2

(0+5, - 4+2) or (5, -2) is the second point.

45

2 xy

(0, -4)

(5, -2)

2.5

04/21/23 Math 120 - KM 34

Graph

(0, 4)

(2, 3)

42

1 xy

(0, 4) is the y-intercept

-1/2 is the slopeRun 2 then rise -1

(0+2, 4+-1) or (2, 3) is the second point.

2.5

04/21/23 Math 120 - KM 35

Graph

(0, 0)

(3, -2)

032 yx

xy3

2Solve for y

(0, 0) is the y-intercept

-2/3 is the slopeRun 3 then rise -2

(0 + 3, 0 + -2) or (3, -2) is the second point.

2.5

04/21/23 Math 120 - KM 36

Uh oh, no y !

(-4, 0)

082 x4x

The x-coordinate is always - 4,so this is a vertical line! Plot (-4,

#)

2.5

04/21/23 Math 120 - KM 37

Uh Oh, no x?

(0, 2)

1882 y

2yThe y-coordinate is always 2,

so this is a horizontal line! Plot (#, 2)

2.5

04/21/23 Math 120 - KM 38

Special Pairs of Lines

2.5

04/21/23 Math 120 - KM 39

Parallel Lines

Are in the same plane And

never intersect!

Parallel Lines have the same

slope!

2.5

04/21/23 Math 120 - KM 40

Perpendicular Lines

Intersect to form a right angle

(90 degrees)!

Perpendicular lines have slopes that are

negative reciprocalsThe slopes multiply to -1.(as long as they are not vertical or horizontal.)

2.5

04/21/23 Math 120 - KM 41

m m|| m

What’s My Slope?

73

2 xy

3

2

3

2

2

3

73

2 xy

2.5

04/21/23 Math 120 - KM 42

m m|| m

What’s My Slope?1243 yx

4

3

3

4

4

3

34

3 xy

2.5

04/21/23 Math 120 - KM 43

m m|| m

What’s My Slope?0 yx

1 11

xy 1

2.5

04/21/23 Math 120 - KM 44

m m|| m

What’s My Slope?

14y

0 UND0

14y

2.5

04/21/23 Math 120 - KM 45

m m|| m

What’s My Slope?

03 x

UND 0UND

03 x

2.5

04/21/23 Math 120 - KM 46

Solve this Mystery!

632 yx

Are these lines parallel

764 yx

23

2 xy

6

7

3

2 xy

2.5

04/21/23 Math 120 - KM 47

Solve this Mystery!

632 yx

Are these lines parallel

1246 yx

23

2 xy

32

3 xy

2.5

04/21/23 Math 120 - KM 48

OK … here it comes!

Determine the equation of the line through the point (4,1) that is perpendicular

to y = -3x+7

3m3

1m

)x()(y 43

11

2.5

04/21/23 Math 120 - KM 49

2.6 Finding Equations of Lines;

Applications

2.6

04/21/23 Math 120 - KM 50

Linear Equations

CByAx

bmxy

)xx(myy 11

2.6

04/21/23 Math 120 - KM 51

What’s my line?

bmxy

)xx(myy 11

3

2m ),(P 15

Clues!

)x()(y 53

21

2.6

04/21/23 Math 120 - KM 52

Look Closely!

bmxy

)xx(myy 11

4

3m ),(P 50

Aha!

54

3

xy

2.6

04/21/23 Math 120 - KM 53

Where’s m ?

12

12

xxyy

m

)xx(myy 11

),(P 861

Uh oh!

),(P 742

)( 64

87

2

1

))(x(y 62

18

2.6

04/21/23 Math 120 - KM 54

Look !

),(P 1731 ),(P 1742

17y

2.6

04/21/23 Math 120 - KM 55

I bet I can’t fool you !

),(P 1731 ),(P 032

3x

2.6

04/21/23 Math 120 - KM 56

A Chemistry Problem?

In the Fahrenheit system, water freezes at 32oF

and boils at 212oF.In the Celsius system, water

freezes at 0oC and boils at 100oC.

Using (F, C) , (32o , 0o), and (212o, 100o)

determine a formula to calculate C if F is known!

2.6

04/21/23 Math 120 - KM 57

A Chemistry Problem - clues?

(F, C) (32o , 0o) (212o, 100o)

)xx(myy 11

)FF(mCC 11

9

5

180

100

32212

0100

m

2.6

04/21/23 Math 120 - KM 58

A Chemistry Problem - clues?

(F, C) (32o , 0o) (212o, 100o)

)FF(mCC 11

(F1, C1)

)F(C 329

50

9

160

9

5 FC

2.6

04/21/23 Math 120 - KM 59

Up in the Air?

An ultralight crop duster is climbing at a rate of 800

ft/min.

Write an equation which gives the height of the

plane in terms of the time after take off.

What is the height 3 minutes after take off?

2.6

04/21/23 Math 120 - KM 60

Up in the Air – Clues!

An ultralight crop duster is climbing at a rate of 800

ft/min.

2.6

04/21/23 Math 120 - KM 61


The crop duster takes off from ground level. So his height at

the start is 0 feet.

2.6

04/21/23 Math 120 - KM 62


What is the height 3 minutes after take off? t = 3

2.6

04/21/23 Math 120 - KM 63

That’s All For Now!

04/21/23 Math 120 - KM 64

Where We Left Off Last Class

12/7/2015math 120 - km1 chapter 3 3.1 cartesian coordinates3.1 3.2 linear functions: graphs and...

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