Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.4 Equations of Lines and Linear...

Copyright copy 2013 2009 2005 Pearson Education Inc

Section 24

Equations of Lines and Linear

Models


Objectives

bull Point-Slope Form

bull Horizontal and Vertical Lines

bull Parallel and Perpendicular Lines


0 and 1xf x a a a The line with slope m passing through the point (x1 y1) is given by

y = m(x ndash x1) + y1

or equivalently

y ndash y1 = m(x ndash x1)

the point-slope form of a line

POINT-SLOPE FORM


Example

Find the point-slope form of a line passing through the point (3 1) with slope 2 Does the point (4 3) lie on this line SolutionLet m = 2 and (x1 y1) = (31) in the point-slope form

To determine whether the point (4 3) lies on the line substitute 4 for x and 3 for y


y minus 1 = 2(x ndash 3)

3 ndash 1 2(4 ndash 3)

2 = 2The point (4 3) lies on the line because it satisfies the point-slope form


Example

Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope

2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


Example (cont)

We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following

If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes

y ndash y1= m(x ndash x1)

1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example

Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1

2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)

Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b

1 4( )2 b

2 4 b

6 bThe slope-intercept form is y = minusx + 6


Example

Find a line y = mx + b that models the data in the table

SolutionCarefully make a scatter plot of the data and sketch a line

x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)


SolutionChose two points on the line and find the equation

The equation of the line is y = 6(x ndash 3) + 18 or y = 6x

x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m


0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b

The equation of a vertical line with x-intercept h is x = h

EQUATIONS OF HORIZONTAL AND

VERTICAL LINES


Example

Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)

Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)


Example (cont)

Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2


0 and 1xf x a a a

Two lines with the same slope are parallel

Two nonvertical parallel lines have the same slope

PARALLEL LINES


Example

Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)

SolutionThe line has a slope of 3 any parallel line also has slope 3

1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a

Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular

A vertical and horizontal line are perpendicular

PERPENDICULAR LINES


Example

Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)

SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows

1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example

Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is

y2

y1

1

3 1

2 2y x


Example (cont)

Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is

y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Objectives

bull Point-Slope Form

bull Horizontal and Vertical Lines

bull Parallel and Perpendicular Lines




or equivalently



POINT-SLOPE FORM


Example








Example


2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


Example (cont)




1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m




or equivalently



POINT-SLOPE FORM


Example








Example


2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


Example (cont)




1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example








Example


2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


Example (cont)




1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example


2 1

2 1

y ym

x x

5 3

2 2

2

4

1

2


Example (cont)




1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example (cont)




1)3 ( )

2( 2y x

13 ( 2)

2y x

15 ( 2)

2y x


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example


2 1

y ym

x x

5 2

1 4

1

3

3


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example (cont)


1 4( )2 b

2 4 b



Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example



x 1 2 3 4 5

f(x) 6 11 18 24 31

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example (cont)




x 1 2 3 4 5

f(x) 6 11 18 24 31

24 18

4 3m

6m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m





VERTICAL LINES


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example




Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example (cont)



0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


0 and 1xf x a a a



PARALLEL LINES


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example



1 1( )y m x x y

3( 2) 1y x 3 6 1y x

3 5y x


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


0 and 1xf x a a a



PERPENDICULAR LINES


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example



1 1( )y m x x y

1( 4) 6y x 4 6y x

10y x


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example


y2

y1

1

3 1

2 2y x


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m


Example (cont)


y2

y1

2

2 1

3 3y x

1 2

3 21

2 3m m

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.4 Equations of Lines and Linear...

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Transcript of Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.4 Equations of Lines and Linear...