The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right.

FINDING THE SLOPE OF A L INE

x

y

1 3 5 7 9-1

- 1

9

7

5

3

1Two points on a line are all that is needed to find its slope.

slope = =5 - 2 3

=2 - 0 2

FINDING THE SLOPE OF A L INE

The slanted line at the right rises 3 units for each 2 units of horizontalchange from left to right. So, the slope m of the line is .3

2

The slope m of a nonvertical line isthe number of units the line rises orfalls for each unit of horizontal change from left to right.

nonvertical line

(2, 5)

rise = 5 - 2 = 3 units

riserun

run = 2 - 0 = 2 units

(0, 2)

x

y

run change in x

The slope m of the nonvertical linepassing through the points (x1, y1) and (x2, y2) is

Read y1 as “y sub one”Read x1 as “x sub one”

m = =

(x1, y1)

(x1, y1)(x2, y2)

(x2, y2)

y2 - y1

(y2 - y1 )

x2 - x1

(x2 - x1 )

FINDING THE SLOPE OF A L INE

(x2, y2)(x1, y1)

rise change in y =

m = run change in xrise change in y = = y2 - y1

x2 - x1

y2 - y1m =x2 - x1

The order of subtraction is important. You can label either point as (x1, y1) and the other point as (x2, y2). However, both the numerator and denominator must use the same order.

numerator

y2 - y1

denominator

x2 - x1

FINDING THE SLOPE OF A L INE

When you use the formula for the slope,

Subtraction order is the same

the numerator and denominator must use the same subtraction order.

CORRECT

x1 - x2

y2 - y1 Subtraction order is differentINCORRECT

Let (x1, y1) = (-1, 2) and(x2, y2) = (3, 2)(3, 2)(x2, y2)

(-1, 2)(x1, y1)

Find the slope of a line passing through (-1, 2) and (3, 2).

Slope is zero. Line is horizontal.

x

y

1 3 5 7 9-1

-1

9

7

5

3

1

line

SOLUTION

m =

run = 3 - ( -1) = 4 units3 - (-1)

rise = 2 - 2 = 0 units

2 - 2 = Substitute values.

= 0

A Line with a Zero Slope is Horizontal

x2 - x1 Run: difference of x-valuesRise: difference of y-valuesy2 - y1

Simplify.= 04

(-1, 2)

(-1, 2)

(3, 2).

(3, 2)

(-1, 2)

(3, 2).

(3, 2)

(x1, y1) (-1, 2)(x2, y2)

x

y

5 15 25 35 45

2700

2500

2300

2100

1900

Hei

ght (

feet

)

Time (seconds)

You are parachuting. At time t = 0 seconds, you open your parachute at h = 2500 feetabove the ground. At t = 35 seconds, you are at h = 2115 feet.

a. What is your rate of change in height?

b. About when will you reach the ground?

Slope as a Rate of Change

INTERPRETING SLOPE AS A RATE OF CHANGE

(0, 2500)

t = 0 seconds, h = 2500 feett = 35 seconds, h = 2115 feet.

t = 0 seconds, h = 2500 feett = 35 seconds, h = 2115 feet.

(35, 2115)

SOLUTIONa. Use the formula for slope to find the rate of change. The change in time is 35 - 0 = 35 seconds. Subtract in the same order. The change in height is 2115 - 2500 = -385 feet.

Rate of Change =

rate of change.

Change in Time

change in time

Rate of Change = m (ft/sec)

Change in Height = - 385 (ft)

m

m =

- 385

- 385

Change in Time = 35 (sec) 35

35= -11

Slope as a Rate of Change

VERBAL MODEL

Your rate of change is -11 ft/sec. The negative value indicates you are falling.

Change in Height

change in height

LABELS

ALGEBRAIC MODEL

b. Falling at a rate of -ll ft/sec, find the time it will take you to fall 2500 ft.

Time =Distance

RateRate

-11 ft/sec

You will reach the ground about 227 seconds after opening your parachute.

Time

Time

Slope as a Rate of Change

Distance

-2500 ft= 227 sec