section 8.1 the slope of a line. definition- linear equation any equation that can be put into the...

Section 8.1The Slope of a Line

Definition- Linear Equation

• Any equation that can be put into the from ax + by = c, where a, b, and c are real numbers and a and b are not both 0, is called a linear equation in two variables.

• The graph will be a straight line.

• The form ax + by = c is called standard form.

Standard Form

• y = 3x + 4• 5 – x = 2y

• -3x + y = 4• -x – 2y = -5

or • x + 2y = 5

Slope

Slope describes the direction of a line.

Guard against 0 in the

denominator

SlopeIf x1 x2, the slope of the line

through the distinct points (x1, y1) and (x2, y2) is

12

12

xx

yy

xinchange

yinchange

run

risem

Why is this

needed?

x-axis

y-axis

Find the slope between (-3, 6) and (5, 2)

RiseRun

-48

-12

= =

Find the slope between (-3, 6) and (5, 2)

12

12

xx

yym

)3()5(

)6()2(

m8

4

2

1

Find the Slope

(5, -2)

(11, 2)

(3, 9)

12

12

xx

yym

311

921

m

Yellow

511

)2(22

m

Blue

35

923

m

Red

8

7

3

2

2

11

Find the slope between (5, 4) and (5, 2).

12

12

xx

yym

)5()5(

)4()2(

m0

2 STOP

This slope is undefined.

x-axis

y-axis

Find the slope between (5, 4) and (5, 2).

RiseRun

-20

Undefined= =

Find the slope between (5, 4) and (-3, 4).

12

12

xx

yym

)5()3(

)4()4(

m8

0

This slope is zero.

x-axis

y-axis

Find the slope between (5, 4) and (-3, 4).

RiseRun

0-8

Zero= =

From these results we can see...

•The slope of a vertical line is undefined.

•The slope of a horizontal line is 0.

Find the slope of the line 4x - y = 8

)0()2(

)8()0(

m2

8

Let x = 0 to find the

y-intercept.

8

8

8)0(4

y

y

y Let y = 0 to find the

x-intercept.

2

84

8)0(4

x

x

x

(0, -8) (2, 0)

4

Find the slope of the line 4x - y = 8 Here is an easier way

Solve for y.

84 yx

84 xy

84 xy

When the equation is solved for y the coefficient

of the x is the slope. m

m = 4

x-axis

y-axis

Graph the line that goes through (1, -3) with (1,-3)

4

3m

Sign of the Slope

Which have a positive slope?

GreenBlue

Which have a negative slope?

RedLight BlueWhite

Undefined

ZeroSlope

Slope of Parallel Lines

• Two nonvertical lines with the same slope are parallel.

• Two nonvertical parallel lines have the same slope.

Are the two lines L1, through (-2, 1) and (4, 5) and L2, through (3, 0)

and (0, -2), parallel?

)0()3(

)2()0(2

m

)2()4(

)1()5(1

m

6

4

3

2

3

2

21

21

LL

mm

x-axis

y-axis

Perpendicular Slopes4

31

m

3

42 m 4

3

What can we say about the intersection of the two white lines?

Slopes of Perpendicular Lines

• If neither line is vertical then the slopes of perpendicular lines are negative reciprocals.

• Lines with slopes that are negative reciprocals are perpendicular.

• If the product of the slopes of two lines is -1 then the lines are perpendicular.

• Horizontal lines are perpendicular to vertical lines.

Write parallel perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3).

)5()3(

)9()7(1

m

)0()8(

)2()3(2

m

2

16

8

8

1

1

8

8

18

8 1

21

21 1

LL

mm

Section 8.2The Equation of a Line

Page 482

Objectives

• Write the equation of a line, given its slope and a point on the line.

• Write the equation of a line, given two points on the line.

• Write the equation of a line given its slope and y-intercept.

Objectives

• Find the slope and the y-intercept of a line, given its equation.

• Write the equation of a line parallel or perpendicular to a given line through a given point.

• Apply concepts of linear equations to realistic examples.

Point-slope Form

mxx

yy

12

12 12 xx 12 xx

1212 xxmyy

Objective Write the equation of a line, given its slope and a point on the line.

Margin 1bThrough (-2, 7); m = 3

11 xxmyy

• These loose there subscripts and become generic variables.

• These variables represent specific values. This is where you substitute.

Margin 1aThrough (-2, 7); m = 3

11 xxmyy

237 xy

637 xyyx 313

133 yx 133 yx

Horizontal and Vertical Lines

• If k is a constant, the vertical line though (k, y) has equation x = k.

• If k is a constant, the horizontal line though ( x, k,) has equation y = k.

Write the equation of the line through (8, -2); m = 0

11 xxmyy

802 xy02 y

2y

Write an equation in standard form through (-1, 2) and (5, 7).

First calculate the slope.

12

12

xx

yym

)1(5

27

6

5

Now plug into point slope. 11 xxmyy

56

57 xy

5576 xy

255426 xy

1765 yx

Slope-intercept Form

• Suppose we have a line with slope m.

• Then the intercept occurs at some point

(0, b)

11 xxmyy

0 xmby

mxby bmxy

Write an equation in standard form with m = 2 and passing

through (0, -3).

bmxy

32 xy

32 yx

32 yx

Find the slope and the

y-intercept of 2x - 5y = 1

bmxy Solve for y and then we will be able to read the answer.

152 yx

yx 512

yx5

5

5

1

5

2

5

1

5

2 xy

5

2m

5

1,0

Write an equation in standard form for the line through (5, 7) parallel to

2x - 5y = 15.• For lines to be

parallel they must have the same slope.

• Solve for y

• Plug into point slope.

bmxy 11 xxmyy

CByAx

standard form

5

2m


2x - 5y = 15.

bmxy 1552 yx

1525 xy

5

15

5

2

5

5

xy

35

2 xy

xx 22


2x - 5y = 15.

11 xxmyy

We know the slope and we know a point.

)7,5(

5

2m

55

27 xy

55

27 xy

• Now we have to change to standard form.

CByAx

555

275 xy

102355 xy352352 xx

2552 yx


2x - 5y = 15.

35

2 xy

55

2 xy

Write an equation in standard form for the line through (-8, 3) perpendicular to

2x - 3y = 10.

• For lines to be perpendicular they must have negative reciprocals of each other.

• Solve for y to find m1

• Identify m2

• Plug into point slope.

bmxy

11 xxmyy

CByAx

standard form

112 )( mm

3

21 m


2x - 3y = 10.

bmxy 1032 yx

1023 xy

3

10

3

2

3

3

xy

3

10

3

2 xy

xx 22

Identify the slope of the perpendicular.

• The slope of the perpendicular line is the negative reciprocal of m1

• Take the negative reciprocal of m1

• Flip it over and change the sign.

2

3

3

2)(

11

12

mm


2x - 3y = 10.

11 xxmyy

We know the perpendicular slope and we know a point.

)3,8(

2

32

m

)8(2

33

xy

)8(2

33

xy

• Now we have to change to standard form.

CByAx

282

332

xy

24362 xy6363 xx

1823 yx


2x - 3y = 10.

3

10

3

2 xy

92

3

xy

Summary of Forms

CByAx • Standard form

• Slope =

B

Am

0,A

C

B

C,0

• x-intercept

• y-intercept

Summary of Forms

• Vertical line– Slope is undefined

– x-intercept is (k, 0)

• Horizontal line– Slope is 0.

– y-intercept is (0, k)

kx

ky

Summary of Forms

bmxy

• Slope-intercept form

• Slope is m.

• y-intercept is (0, b)

Summary of Forms

)( 11 xxmyy

• Point-slope form

• Slope is m.

• Line passes through (x1, y1)

8.2 HomeworkPage 489

1-51 odd

Section 8.4Introduction to Functions

Page 500

Relation• A relation is a set of ordered pairs of real

numbers.

F = {(3, 2) (4, 1) (2, 4) (1, 3)}

• If I say (2, __ ) , can you fill in the blank?

G = {(3, 3) (4, 1) (2, 1) (1, 3)}

• If I say (4, __ ) , can you fill in the blank?

• In a domain the set of all of the values of the independent variable is called the domain.

• What is the domain of F?

{3, 4, 2, 1}

• Does G = {(3, 3) (4, 1) (2, 1) (1, 3)} have the same domain?

DomainF = {(3, 2) (4, 1) (2, 4) (1, 3)}

• In a domain the set of all of the values of the dependent variable is called the range.

• What is the range of G?

{3, 1}

• Does F = {(3, 2) (4, 1) (2, 4) (1, 3)} have the same range?

Range G = {(3, 3) (4, 1) (2, 1) (1, 3)}

(Domain, Range)

• Notice the alphabetical characteristic of Domain and Range.

(x, y)

(a, b)

(abscissa, ordinate)

• Unfortunately (independent, dependent) breaks the rule.

Function

• A function is a relation in which , for each value of the first component there is exactly one value of the second component.

H = {(3, 2) (4, 1) (3, 4) (1, 3)}

K = {(2, 3) (4, 1) (3, 1) (2, 3)}

• H is not a function,but K is a function.

Function Expressed as a Mapping

A

C

B

1

2

3

F =

{(A,1)

(C, 2)

(B, 3)}

Domain Range

Since A goes to two ranges G is not a function.

Function Expressed as a Mapping

A

C

B

1

2

3

G =

{(A,1)

(C, 2)

(B, 3)

(A, 4)}

Domain Range

4

Finding Domains and Ranges from Graphs

-4 4

6

-6

The range runs from -6 to 6

[-6, 6]

The domain runs from -4 to 4

[-4, 4]


The range runs from - to

(-, )

The domain runs from - to

(-, )


The range runs from -3 to

[-3, )

The domain runs from - to

(-, )-3

Identify a Function from an Equation

First consider the domain

• Exactly what is the domain? (what can x be?)

• For any choice of x in the domain there should be exactly one value in the range.

32 xyDomain(-, ) Over the entire domain there is

no choice for x that corresponds to two values of y.

Identify a Function from an Equation

Whenever there is a fraction you must guard against a zero denominator.

1

5

xy

Domain

,11,

,

Range01xdisasterx 1

Vertical Line Test

If a vertical line intersects the graph of a relation in more than one point, then the relation is not a function.

H = {(3, 2) (4, 1) (3, 4) (1, 3)}

Two intersectionstherefore not a function

Vertical Line Test

Vertical Line Test

• Think of the uppercase letters. ABCDEFGHIJKLMNOP...

• Which of the uppercase letters would pass the vertical line test?

Functional Notation

• y = F(x)

• F(x) read F of x

• It does not mean F × x (multiplication)

Functional Notation

• Consider y = 2x + 5• Suppose that you wanted to tell someone to

substitute in x = 3 into an equation.

• With functional notation y = 2x + 5 becomes f(x) = 2x + 5.

• And f(3) means substitute in 3 everyplace you see an x.

Linear Function

A function that can be written in the form

f(x) = mx + bis a linear function.

Domain

,

Range

,

Section 8.5 Function Notation

Page 513

Functional Notation

• y = F(x)

• F(x) read F of x

• It does not mean F × x (multiplication)

Functional Notation

• Consider y = 2x + 5• Suppose that you wanted to tell someone to

substitute in x = 3 into an equation.

• With functional notation y = 2x + 5 becomes f(x) = 2x + 5.

• And f(3) means substitute in 3 everyplace you see an x.

Linear Function

A function that can be written in the form

f(x) = mx + bis a linear function.

Domain

,

Range

,

Example 1 If f(x) = 7.5x, find f(0), f(10), f(20).

f(0) = 7.5(0) = 0

f(10) = 7.5(10) = 75

f(20) = 7.5(20) = 150

Example 5 Find f(0), f(3), and f(-2) 2( ) 3 2 1f x x x

2(0) 3(0) 2(0) 1f

(0) 0 0 1f

(0) 1f

Example 5 Find f(0), f(3), and f(-2) 2( ) 3 2 1f x x x

2(3) 3(3) 2(3) 1f (3) 3(9) 2(3) 1f

(3) 27 6 1f (3) 32f

Example 5 Find f(0), f(3), and f(-2) 2( ) 3 2 1f x x x 2( 2) 3( 2) 2( 2) 1f

( 2) 3(4) 2( 2) 1f

( 2) 12 4 1f ( 2) 7f

Homework problem 4 g(-2)2( ) 3 4g x x x

2( 2) ( 2) 3( 2) 4g ( 2) 4 3( 2) 4g ( 2) 4 ( 6) 4g ( 2) 2g

Homework problem #10 f(2) - g(3) 2( ) 3 4g x x x 2( ) 3 2 1f x x x

2(2) 3(2) 2(2) 1f

(2) 3(4) 2(2) 1f

(2) 12 4 1f (2) 15f

2(3) (3) 3(3) 4g

(3) 9 3(3) 4g

(3) 9 9 4g

(3) 22g

(2) (3)f g 15 22 7

#47 Graph the function Draw x = 4Draw f(4)

1( ) 2

2f x x

x f(x)

0 2

2 3

4 4

(4) 4f

Homework 8.51 – 51

Odd or EOO

Section 8.6 Algebra with Functions

Page 522

Example 1If

and

then write the formula for the functions for

2( ) 4 3 2f x x x 2( ) 2 5 6g x x x

, , , /f g f g fg f g

Example 12( ) 4 3 2f x x x 2( ) 2 5 6g x x x f g

2(4 3 2)x x 2(2 5 6)x x 2 24 3 2 2 5 6x x x x

26 2 4x x

Example 12( ) 4 3 2f x x x 2( ) 2 5 6g x x x f g

2(4 3 2)x x 2(2 5 6)x x 2 24 3 2 2 5 6x x x x

22 8 8x x

Example 12( ) 4 3 2f x x x 2( ) 2 5 6g x x x fg

2(4 3 2)x x 2(2 5 6)x x

4 3 2 3 2 28 20 24 6 15 18 4 10 12x x x x x x x x

4 3 28 14 35 28 12x x x x

fg x

Example 12( ) 4 3 2f x x x 2( ) 2 5 6g x x x /f g

2(4 3 2)x x 2(2 5 6)x x

fx

g

Composition Example If

and

then write the formula for the functions for

( ) 5f x x 2( ) 2g x x x

,f g x g f x

Composition Example

( ) 5f x x 2( ) 2g x x x f g x

( )( ) 5f g x

2 2 5x x

2 2 5f g x x x

Composition Example

( ) 5f x x 2( ) 2g x x x g f x

2( )( ) 2g f x

25 2 5x x

2 10 25 2 10x x x 2( )( ) 8 15g f x x x

Homework1 – 43

Odd or EOO

section 8.1 the slope of a line. definition- linear equation any equation that can be put into the...

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