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.

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F INDING THE S LOPE OF A L INE. 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. F INDING THE S LOPE OF A L INE. nonvertical line. y. 9 7 5 3 1. The slanted line at the right rises

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FINDING THE SLOPE OF A LINE

The slopem 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 LINE

nonvertical line

y

9

7

5

3

1

The slanted line at the right rises

3 units for each 2 units of horizontal

change from left to right. So, the

slope m of the line is .

(2, 5)

3

2

rise = 5 - 2 = 3 units

(0, 2)

run = 2 - 0 = 2 units

- 1

1 3 5 7 9

x

-1

rise

run

5 - 2 3

=

The slopem of a nonvertical line is

the number of units the line rises or

falls for each unit of horizontal change from left to right.

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

slope =

=

2 - 0 2

FINDING THE SLOPE OF A LINE

(x1,y1)

(x1,y1)

y

(x2,y2)

(x2,y2)

(x2, y2)

(x1, y1)

(y2 - y1)

(x2 - x1)

x

rise change in y

=

x2-x1

run change in x

y2-y1

The slope m of the nonvertical line

passing through the points(x1, y1)

and(x2, y2)is

Read y1 as “y sub one”

Read x1 as “x sub one”

m =

=

FINDING THE SLOPE OF A LINE

y2-y1

m =

x2-x1

rise change in y

y2-y1

m =

=

=

run change in x

x2-x1

the numerator and denominator

must use the same subtraction order.

numerator

denominator

y2-y1

CORRECT

Subtraction order is the same

x2-x1

y2-y1

INCORRECT

Subtraction order is different

x1-x2

When you use the formula for the slope,

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.

A Line with a Zero Slope is Horizontal

line

(3,2).

(3,2).

(-1,2)

(-1, 2)

y

(-1,2)

9

7

5

3

1

(x1,y1)

Let (x1, y1) = (-1, 2) and

(x1,y1)

(-1,2)

(x2, y2) = (3, 2)

(x2,y2)

(x2,y2)

(3,2)

(3,2)

y2 - y1

Rise: difference of y-values

rise = 2 - 2 = 0 units

x2 - x1

Run: difference of x-values

(-1,2)

(3,2)

2 - 2

=

3 - (-1)

run = 3 - ( -1) = 4 units

0

4

Simplify.

=

-1

1 3 5 7 9

x

-1

=

0

Slope is zero. Line is horizontal.

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

SOLUTION

m=

Substitute values.

Slope as a Rate of Change

(0,2500)

h = 2500 feet

h = 2500 feet

t = 0 seconds,

t = 0 seconds,

t = 35 seconds,

t = 35 seconds,

h = 2115 feet.

h = 2115 feet.

y

2700

2500

2300

2100

1900

(35,2115)

Height (feet)

x

5 15 25 35 45

Time (seconds)

INTERPRETING SLOPE AS A RATE OF CHANGE

You are parachuting. At time t = 0 seconds, you open your parachute at h = 2500 feet

above 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

rate of change.

change in time

change in height

Change in Height

Rate of

Change

=

Change in Time

m

- 385

LABELS

35

- 385

ALGEBRAIC

MODEL

=

=

m

35

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

SOLUTION

a. 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.

VERBAL

MODEL

Rate of Change = m(ft/sec)

Change in Height =- 385 (ft)

Change in Time = 35 (sec)

-11

Slope as a Rate of Change

Distance

=

Time

Time

Distance

Rate

Time

-2500 ft

=

227 sec

-11 ft/sec

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

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

Rate