copyright © 2013, 2009, 2005 pearson education, inc. section 2.3 the slope of a line

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 2.3

The Slope of a Line


Objectives

• Slope

• Slope-Intercept Form of a Line

• Interpreting Slope in Applications


, 0 and 1,xf x a a a

The slope of the line passing through the points (x1, y1) and (x2, y2) is

where x1 ≠ x2. That is, slope equals rise over run.

SLOPE

2 1

2 1

,y y

mx x


Example

Find the slope of the line passing through the points (7, 3) and (2, 2). Plot these points and graph the line. Interpret the slope. Solution

2 1

2 1x

ym

y

x

2

2

7

3

1

5

5

1

Graph the line passing through these points. The slope indicates that the line rises 1 unit for every 5 units of run.


Slope

A line with positive slope rises from left to right. A line with negative slope falls from left to right. A horizontal line has a zero slope. A line with undefined slope is a vertical line.


Example

Find the slope of the line passing through each pair of points, if possible. a. (3, 4), (−1, 4) b. (−4, 2), (−4, 5) Solutiona.

b.

2 1

2 1

y ym

x x

4 4

1 3

0

4

0

2 1

2 1

y ym

x x

5 2

4 ( 4)

3

0 undefined


Example

Sketch a line passing through the point (1, 2) and having slope 3/4.

SolutionStart by plotting (1, 2).The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4.The line passes through the point (1 + 4, 2 + 3) = (5, 5).


, 0 and 1,xf x a a a The line with slope m and y-intercept b is given by

y = mx + b,

the slope-intercept form of a line.

SLOPE-INTERCEPT FORM


Example

For the graph write the slope-intercept form of the line.

Solution

The graph intersects the y-axis at 0, so the y-intercept is 0.

The graph falls 3 units for each 1 unit increase in x, the slope is –3.

The slope intercept-form of the line is y = –3x .


Example

For the graph shown, write the slope-intercept form of the line. Solution

The graph passes through (0, 1), so the y-intercept is 1. Because the graph rises 5 units for each unit increased in x, the slope is 5. The slope-intercept form is y = 5x + 1.


Example

The points listed on the table all lie on a line.a. Find the missing value in the table.b. Write the slope-intercept form of the line.Solutiona. The line passes through (−4, 9) and (2, −3).

For each unit increased in x, y decreases by 2. Therefore the missing value is 5.

b. The slope-intercept form is y = −2x + 1.

x y

−4 9

2 −3

? −9

2 ( )

3

4

9m

2

6

1

2


Example

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept represent?

Solutiona. The y-intercept is 35, so the boat is initially 35 miles from the dock.


Example (cont)

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.b. The graph passes through the point (4, 15). Discuss the meaning of this point.

Solutionb. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock.


Example (cont)

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.c. Find the slope of the line. Interpret the slope as a rate of change.

Solution

c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

15 05

4 7m


Example

When a street vendor sells 40 tacos, his profit is $24, and when he sells 75 tacos, his profit is $66.a. Find the slope of the line passing through the points (40, 24) and (75, 66)b. Interpret the slope as a rate of change.Solution

b. Profit increases on average, by $1.20 for each additional taco sold.

66 24 42a. 1.2

75 40 35m

copyright © 2013, 2009, 2005 pearson education, inc. section 2.3 the slope of a line

Documents

slope slide

line interpreting slope

line slide

slope interceptform

undefined slope

slopeintercept form

positive slope

negative slope