**The Slope of a Line**

Let’s go for a ride on the Gainesville-Hawthorne Trail. We’ll start at Boulware Springs (Mile 0) and ride out to the Lochloosa Trailhead (Mile 15) and back.

https://www.floridastateparks.org/trail/Gainesville-Hawthorne

Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.

Let's make a numerical representation to get started.

Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.

Initially we are at Mile 0.

Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.

After 1 hour we reach Mile 15.

Fifteen minutes (0.25 hours) later we are still at Mile 15.

Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.

Finally, after a total of 2 hours, we arrive back at Mile 0.

Suppose it takes us 1 hour to get to Hawthorne but only 45 minutes to return after taking a 15-minute break.

From the table we may create a line graph.

The first line segment rises from left to right.

The second is horizontal.

The third falls

from left to right.

Each line segment has a different "tilt." To quantify the tilt we use the concept of

slope

, which is the

ratio of rise over run

.

The increasing line segment rises 15 miles for 1 hour of run so its

slope

is

15 miles per hour

.

15 miles

1 hour

The horizontal line segment does not rise or fall so its

slope

is

0 miles per hour

.

The decreasing line segment falls 15 miles for 3/4 hour of run so its

slope

is (-15 miles)/(3/4 hour) =

-20 miles per hour

.

-15 miles

3/4 hour

15 mph

We may interpret the

slope

as a

rate of change

, the velocity at which we cycled.

0 mph

-20 mph

In this lesson we will learn more about slope and how it relates to the graph of a linear function.

rise

run

The graph of a linear function is a line.

The "tilt" of the line is called the slope and equals

rise

over

run

.

rise

=

y

₂ -

y

₁

run

=

x

₂ -

x

₁

Slope

(

x

₁,

y

₁)

y

₂ -

y

₁

x

₂ -

x

₁

slope =

(

x₂

,

y₂

)

Slope Man

A

positive slope

indicates that the line rises from left to right.

A

negative slope

indicates that the line falls from

left to right.

This formula is equivalent to

y

=

mx

+

b

where

m

is the slope

and

b

is the

y

-intercept

.

In a previous lesson we learned that the formula for a linear function is

f

(

x

) =

ax

+

b

where

a

is the average rate of change

and

b

is the beginning value

.

f

(

x

) =

a

x

+

b

y

=

m

x

+

b

A horizontal line has

0 slope

.

0

0

A vertical line has

undefined slope

.

The graph of a linear function is a

line

.

y

=

mx

+

b

(0,

b

)

increasing

m

>0

1

m

Plot the

y

-intercept and then use the slope to find other points on the line.

y

=

mx

+

b

y

=

m

x

+

b

y

=

mx

+

b

decreasing

m

<0

(0,

b

)

y

=

mx

+

b

y

=

m

x

+

b

1

m

Slope-Intercept Form for a Line

y

=

mx

+

b

m

is the slope

b

is the

y

-intercept

y

=

m

x

+

b

y

=

m

x

+

b

When a linear function is used to model physical quantities, the slope of its graph provides certain information.

The slope of the graph of a linear function indicates the rate at which a quantity is either increasing or decreasing.

Slope as a Rate of Change

From 1981 to 2000, average public college tuition and fees can be modeled by

f

(

x

) =

136

x

+ 772,

where

x

= 1 corresponds to 1981. The slope of the graph of

f

is

m

= 136

and indicates that, on average,

tuition and fees increased by $136 per year between 1981 and 2000

.

From 1987 to 2004 the number of federally insured banks could be approximated by

N

(

t

) =

-358.4

t

+ 13,723,

where

t

= 0 corresponds to 1987. The slope of the graph of

N

is

m

= -358.4

and indicates that, on average,

the number of federally insured banks decreased by approximately 358 per year from 1987 to 2004

.

Now it's your turn!

Try the following exercises and then watch the video solutions to check.

Determine if

f

(

x

) = 6 - 8

x

is a linear function. If it is, write it in the form

f

(

x

) =

ax

+

b

.

Find the slope of the line passing through (-3,7) and (6,-2).

Determine the slope of the line shown in the graph.

Let

f

be a linear function. Find the slope,

x

-intercept, and

y

-intercept of the graph of

f

.

The line graph shows the number of welfare beneficiaries in millions for selected years.

(a) Find the slope of each line segment.

(b) Interpret each slope as a rate of change.

Year

Welfare Beneficiaries (millions)

Starting at home, a driver travels away from home on a straight highway for 2 hours at 60 miles per hour, stops for 1 hour, and then drives home at 40 miles per hour. Sketch a graph that shows the distance between the driver and home.

Slope

can be interpreted as a

rate of change

of a quantity.