Copyright © 2013 Pearson Education, Inc.

Section 3.6

Point-Slope Form

Page 215

Point-Slope Form

The line with slope

m

passing through the point (

x

1 ,

y

1 ) is given by

y – y 1

=

m

(

x

–

x

1 ), or equivalently,

y

=

m

(

x

–

x

1 ) +

y

1 the

point-slope form

of a line.

Example

Page 215 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?

Solution

Let

m

= 2 and (

x

1,

y

1 ) = ( 3 , 1 ) in the point-slope form.

y y

–

y

1 =

m

(

x

− 1 = 2(

x

–

x

– 3) 1 ) on the line, substitute 4 for

x

and 3 for

y

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

Example

Page 216 Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5).

Solution

Before we can apply the point-slope form, we must find the slope.

m



y

2

x

2  

y x

1 1  2  2 4  1 2

Example (cont)

Page 216 We can use either (−2, 3) or (2, 5) for (

x

1 ,

y

1 ) in the point slope form. If we choose (−2, 3), the point-slope form becomes the following.

y

–

y y

1 =

m

(

x

– 1 2 (

x x

1 ) 2 ) )

y

1 2 (

x

 2) If we choose (2, 5), the point-slope form with

x

1

y

1 = 5 becomes

y

1 2 (

x

 2).

= 2 and

Point-Slope Form

Write the point-slope form and then the slope-intercept form of the equation of the line with slope 6 that passes through the point (2,-5).

SOLUTION

y y

  

y

5 1  

m



x x y

 5  6 (

x

 

x

1 2

 

 2 ) Substitute the given values Simplify This is the equation of the line in

point-slope form

.

y

 5

y

  6

x

6

x

  12 17 Distribute Subtract 5 from both sides This is the equation of the line in

slope-intercept form

.

Point-Slope Form

Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (-2,-1) and (-1,-6).

SOLUTION First find the slope of the line. This is done as follows:

m

  1   2  (  1 )  5  1   5 Blitzer, Introductory Algebra, 5e – Slide #7 Section 4.5

Point-Slope Form

CONTINUED Use either point provided. Using (-2,-1).

y



y

  

y

1 

y

 1   5

m



x

  5



x





x



x

1 (  2 )   2

 

Substitute the given values Simplify This is the equation of the line in

point-slope form

.

y

 1   5

x

 10

y

  5

x

 11 Distribute Subtract 1 from both sides This is the equation of the line in

slope-intercept form

.

Example

Page 218 Find the slope-intercept form of the line perpendicular to

y

=

x

– 3, passing through the point (4, 6).

Solution

The line

y

=

x

– 3 has slope

m

1 perpendicular line is

m

2 = 1. The slope of the = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.

y

1(

x

 4)

y y x

10 4

Example

Page 220 similar to Example 7 A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons. a. How fast is the pump removing water?

Solution

The pump removes a total of 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour.

Example (cont)

Page 220 similar to Example 7 b. Find the slope-intercept form of a line that models the amount of water in the pool. Interpret the slope.

The line passes through the points (1,8000) and (4, 2000), so the slope is   2000

y y

–

y

1 =

m

(

x

–

x

1 ) – 8000 =

−

2000(

x

– 1)

y

– 8000 =

−

2000

x

+ 2000

y

=

−

2000

x

+ 10,000 A slope of −2000, means that the pump is removing 2000 gallons per hour.

Example (cont)

Page 220 similar to Example 7 c. Find the

y

-intercept and the

x

-intercept. Interpret each.

The

y

-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the

x

intercept let

y

= 0 in the slope-intercept form.

0 2000

x

 

2000

x



10, 000



10, 000

2000 2000

x x

 10, 000  2000 5 The

x

-intercept of 5 indicates that the pool is emptied after 5 hours.

Example

Page 220 similar to Example 7 d. Sketch the graph of the amount of water in the pool during the first 5 hours.

The

x

-intercept is 5 and the

y

-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000).

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 Y 0 1 2 3 4 Time (hours) 5 6 X

Example (cont)

Page 220 similar to Example 7 e. The point (2, 6000) lies on the graph. Explain its meaning. The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water. 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 Y 0 1 2 3 4 Time (hours) 5 6 X

Group Activity (3.5 on page 214) Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130 , and in 2010 it was $7610 . Note that the known value for 2008 is $6530.

Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope.

m

 change in y change in x  7610  6130 2010  2005  1480 5  296 Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr.

Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

m

 change in y change in x  7610  6130 10  05  1480 5  296 Figure not in book

Group Activity (3.5 on page 214) Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation.

y



y



y

1 6130  

m



x

296



x

 1480  

x

1



2005



5  296

y

This is the equation of the line in

point-slope form

.

 6130

y

  296

x

 593480 296

x

 587350 Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

y y

 

y

1 6130  

m



x

 296



x x

1 



5



This is the equation of the line in

slope-intercept form

.

This is the equation of the line in

point-slope form

.

y

 6130  296

x

 1480

y

 296

x

 4650 This is the equation of the line in

slope-intercept form

.

Group Activity (3.5 on page 214) Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008.

Substitute 2008 or 8 for x and compute y.

y

 296

x

 587350

y

 296 ( 2008 )  587350  7018

y

 296 ( 8 )  4650  7018 The model predicts that the tuition in 2008 will be $7018 and the tuition in 2015 will be $9090.

Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015.

Substitute 2015 or15 for x and compute y.

y

 296 ( 2015 )  587350  9090

y

 296 ( 15 )  4650  9090

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.

Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope.

m

 change in y change in x  35 .

3  30 .

0 30  10  5 .

3  0 .

265 20 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.

(10, 30.0) (30, 35.3)

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.

m

 5 .

3  0 .

265 20 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.

y



y

 30 .

0

y

1  

m



0 .

265

x





x x

1 



10



This is the equation of the line in

point-slope form

.

y

 30 .

0

y

 0  0 .

265

x

.

265

x

  2 .

65 27 .

35 This is the equation of the line in

slope-intercept form

.

A linear equation that models the median age of the U.S. population, y, x years after 1970.

(10, 30.0) (30, 35.3)

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.

m

 5 .

3  0 .

265 20 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.

y

 0 .

265

x

 27 .

35 A linear equation that models the median age of the U.S. population, y, x years after 1970.

Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y.

y

 0 .

265 ( 50 )  27 .

35  40 .

6 The model predicts that the median age of the U.S. population in 2020 will be 40.6.

(10, 30.0) (30, 35.3)

Example 6 Modeling female officers (page 219) In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214.

a) The slope of the line passing through (1995, 690) and (2010.1110) is

m

 1110 2010   690 1995  28 b) The number of female officers increased, on average by about 28 officers per year.

y



y

1 

m



x



x

1 

y

 690  28 

x

 1995  c) Estimate how many female officers there were in 2006.

y y

  690 28    28   2006 690   1995 998 

y OR



y

 28

x

28 ( 2006 )   55170 55170

y

 56168  55170  998 // (1995, 690) (2010, 1110) Write the slope-intercept form of the of the line shown in the graph.

y

 690

y

  28

x

28

x

 55860  55170

DONE

Group Activity (3.5 on page 214) Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

y m

 change in y change in x  7610  6130 10  5  1480 5

y

 

y

1 6130  

m



x

 296



x x

1 



5



 296 This is the equation of the line in

point-slope form

.

y

 6130  296

x

 1480

y

 296

x

 4650 This is the equation of the line in

slope-intercept form

.

y

 296 ( 3 )  6130  7018 The model predicts that the tuition in 2008 will be $7018.

Objectives

• Derivation of Point-Slope Form • Finding Point-Slope Form • Applications