Intermediate Algebra (11th edition)

Slope is the ratio of vertical change, or rise,to horizontal change, or run.A sim-

ple way to remember this is to think, “Slope is rise over run.”

OBJECTIVE 1 Find the slope of a line, given two points on the line.To get a

formal definition of the slope of a line, we designate two different points on the line. To

differentiate between the points, we write them as 1 x 1 , y 12 and 1 x 2 , y 22. See FIGURE 14.

148 CHAPTER 3 Graphs, Linear Equations, and Functions

OBJECTIVES Slope (steepness) is used in many practical ways. The slope of a highway (sometimes

called the grade) is often given as a percent. For example, a 10% slope

means that the highway rises 1 unit for every 10 horizontal units. Stairs and roofs

have slopes too, as shown in FIGURE 13.

Aor

10

100 =

1

10 B

The Slope of a Line

3.2

1 Find the slope of a

line, given two

points on the line.

2 Find the slope of a

line, given an

equation of the line.

3 Graph a line, given

its slope and a point

on the line.

4 Use slopes to

determine whether

two lines are parallel,

perpendicular,

or neither.

5 Solve problems

involving average

rate of change.

Slope

(or grade)

is.

(not to scale)

1

10

1

10

Slope (or

pitch) is^13

1

3

Slope is.

7

9

9

7

.

FIGURE 13

x

y

x 2 – x 1 = x

Change in x (run)

0

(x 1 , y 1 )

yChange in 2 – y 1 = y (x 2 , y 2 )

y (rise)

(x 1 , y 2 )

FIGURE 14

Slope Formula

The slope mof the line through the distinct points and is

m 1 x 1 x 22.

rise

run



change in y

change in x



¢y

¢x



y 2 y 1

x 2 x 1

1 x 1 , y 12 1 x 2 , y 22

As we move along the line in FIGURE 14from to , the y-value

changes (vertically) from to an amount equal to As ychanges from

to the value of xchanges (horizontally) from to by the amount

NOTE The Greek letter delta, ,is used in mathematics to denote “change in,” so

yand xrepresent the change in yand the change in x, respectively.

The ratio of the change in yto the change in x(the rise over the run) is called the

slopeof the line, with the letter mtraditionally used for slope.

¢ ¢

¢

y 2 , x 1 x 2 x 2 - x 1.

y 1 y 2 , y 2 - y 1. y 1

1 x 1 , y 12 1 x 2 , y 22