chapter 3 section 3 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Chapter Chapter 33Section Section 33

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


The Slope of a Line

11

33

22

3.33.33.33.3

Find the slope of a line given two points.Find the slope from the equation of a line.Use slopes to determine whether two lines are parallel, perpendicular, or neither.


An important characteristic of the lines we graphed in Section 3.2 is their slant, or “steepness.”

One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line.

The Slope of a Line

Slide 3.3 - 3


Objective 11

Slide 3.3 - 4

Find the slope of a line given two points.


To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y-values, which is the difference 6 − 1 = 5 units. The horizontal change, or run, is the change in the x-values, which is the difference 5 − 2 = 3 units.

Find the slope of a line given two points.

Slide 3.3 - 5

vertical change in rise

horizontal change in run 3e

5slop

x

y

The slope is the ratio of the vertical change in y to the horizontal change in x.

Count squares on the grid to find the change. Upward and rightward movements are positive. Downward and leftward movements are negative.


EXAMPLE 1 Finding the Slope of a Line

Solution:

Slide 3.3 - 6

Find the slope of the line.

6

1m

6m


The slope of a line can be found through two nonspecific points. This notation is called subscript notation, read x1 as “x-sub-one” and x2 as “x-sub-two”.

The slope of a line is the same for any two points on the line.

Find the slope of a line given two points. (cont’d)

Slide 3.3 - 7

21 2

2 1

1

horizontal change in run

vertical change iif

n ris .

e

y yx x

x x x

ym

Traditionally, the letter m represents slope.

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

Moving along the line from the point (x1, y1) to the point (x2, y2), we see that y changes by y2 − y1 units. This is the vertical change (rise). Similarly, x changes by x2 − x1 units, which is the horizontal change (run). The slope of the line is the ratio of y2 − y1 to x2 − x1.


EXAMPLE 2

Solution:

Finding Slopes of Lines

4 8

2 6m

12

8

3

2

Slide 3.3 - 8

and yield the same slope. Make sure to start with the

x- and y-values of the same point and subtract the x- and y-values of the other point.

Find the slope of the line through (6, −8) and (−2,4).

2 1

2 1x

y y

x 1 2

1 2x

y y

x


Positive and Negative Slopes

A line with a positive slope rises (slants up) from left to right.

A line with a negative slope falls (slants down) from left to right.

Find the slope of a line given two points. (cont’d)

Slide 3.3 - 9

Slopes of Horizontal and Vertical Lines

Horizontal lines, with equations of the form y = k, have slope 0.

Vertical lines, with equations of the form x = k, have undefined slopes.


EXAMPLE 3

Solution:

5 5

1 2m

0

3

0

Slide 3.3 - 10

Find the slope of the line through (2, 5) and (−1,5).

Finding the Slope of a Horizontal Line


EXAMPLE 4

Solution:

Finding the Slope of a Vertical Line

3 3

4 1m

5

0

Slide 3.3 - 11

Find the slope of the line through (3, 1) and (3,−4).

undefined slope


Objective 22

Find the slope from the equation of a line.

Slide 3.3 - 12


3 5y x

7y

3 5y x

11y

Consider the equation y = −3x + 5.

The slope of the line can be found by choosing two different points for value x and then solving for the corresponding values of y. We choose x = −2 and x = 4.

Find the slope from the equation of a line.

Slide 3.3 - 13

3 52y 6 5y

3 4 5y 12 5y

The ordered pairs are (−2,11) and (4, −7). Now we use the slope formula.

11 7 18

63

2 4m


Find the slope from the equation of a line. (cont’d)

Slide 3.3 - 14

Step 1: Solve the equation for y.

Step 2: The slope is given by the coefficient of x.

The slope, −3 is found, which is the same number as the coefficient of x in the given equation y = −3x + 5. It can be shown that this always happens , as long as the equation is solved for y.

This fact is used to find the slope of a line from its equation, by:


EXAMPLE 5

Solution:

Finding Slopes from Equations

33 32 9xyx x

Slide 3.3 - 15

Find the slope of the line 3x + 2y = 9.

3 9

2 2y x

3

2m

2 3

2 2

9y x


Objective 33

Use slopes to determine whether two lines are parallel, perpendicular, or neither.

Slide 3.3 - 16


Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes.

Use slopes to determine whether two lines are parallel, perpendicular, or neither.

Slide 3.3 - 17

1

a

Lines are perpendicular if they intersect at a 90° angle. The product of the slopes of two perpendicular lines, neither of which is vertical, is always −1. This means that the slopes of perpendicular lines are negative (or opposite)

reciprocals—if one slope is the nonzero number a, the other is . The table to the right shows several examples.


EXAMPLE 6Deciding whether Two Lines Are Parallel or Perpendicular

Slide 3.3 - 18

Solution:3 4x y 3 9x y

3 33 4x xyx 3 9xy xx 4 3

1 1

xy

4 3y x

3 9

3 3 3

xy

13

3y x

13 1

3

3m 1

3m

The product of their slopes is −1, so they are perpendicular

Determine whether the pair of lines is parallel, perpendicular, or neither.

chapter 3 section 3 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

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