4-1A Rate of Change and the Slope of a Line Using a Graph

Algebra 1 Glencoe McGraw-Hill Linda Stamper

Rate of change is a ratio that describes, on average, how much one quantity changes with respect to a change in another quantity. If x is the independent variable and y is the dependent variable, then

xx yy

1 6

2 12

3 18

4 24

xinchangeyinchange

changeofrate +6

+6

+6

6

+

1 +

1 +

1

16

The rate of change is 6.

Example 1 Use the table to find the rate of change.

xx yy

2 6

4 12

6 18

8 24

xinchangeyinchange

changeofrate +6

+6

+6

6

+

2 +

2 +

2

2

3

The rate of change is 3.

You have been graphing linear equations which graph as a line.

In this lesson you will learn about slope.

1isslope

Slope is the steepness of a line.

You have been graphing linear equations which graph as a line.

In this lesson you will learn about slope.

Slope is the steepness of a line.

2isslope

You have been graphing linear equations which graph as a line.

In this lesson you will learn about slope.

run horizontal

rise vertical slope Slope is a ratio of rise to run.

31

isslope

Slope is the steepness of a line.

Find the slope of a hill that has a vertical rise of 40 feet and a horizontal run of 200 feet. Let m represent slope.

Write word ratio.runrise

m

Substitute.20040

Simplify – write in lowest terms. 51

Example 2 Find the slope of a wheelchair ramp that has a vertical rise of 2 feet and a horizontal run of 24 feet. Let m represent slope.

runrise

m

242

121

Example 3 Find the slope of a ski trail that has a vertical rise of -3 feet and a horizontal run of 4 feet. Let m represent slope.

runrise

m

34

43

Types of SlopeImagine that you are walking to the right on a

line.A positive slope means that you are walking uphill.

Imagine that you are walking to the right on a line.A negative slope means that you are walking downhill.

Types of Slope

Imagine that you are walking to the right on a line. Zero slope means that you are walking on level ground.

Do not identify this as “no slope”.

Types of Slope

Undefined slope is a vertical line. You can not walk up a vertical line. It is not possible. You would fall!

Do not identify this as “no slope”.

ouch!

Types of Slope

zero

Types of Slope are words!

positive

negative

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Find the Slope by Walking the Line

This means to calculate the slope - to

find the steepness of the line!

Find the Slope by Walking the LineWhen you find the slope by

walking the line, moving upward on the graph is positive. Moving left on the graph is negative.

Moving right on the graph is positive.

x

y

•

•Moving downward on the graph is negative.

Find the slope of the line.

Start at either point. Find the vertical rise. Up is positive and down is negative.

4

Find the horizontal run. Going right is positive and going left is negative.

2

Simplify.

2x

y

•

•

runrise

mWrite slope ratio.Write slope ratio. MANDATORY STEP

Find the slope of the line.

Start at either point. Find the vertical rise. Up is positive and down is negative.

4

Find the horizontal run. Going right is positive and going left is negative.

2

Simplify.

2x

y

•

•

runrise

mWrite slope ratio.

•

••

Slope is the rate of change. It will change -2 all along the line.

Example 4A Find the slope of the line.

2. Start at either point. Find the vertical rise. Up is positive and down is negative.

4

3. Find the horizontal run. Going right is positive and going left is negative.

74

It does not matter at which point you begin the walk!

x

y1. Write slope ratio.runrise

m

•

•

Example 4B Find the slope of the line.

2. Start at either point. Find the vertical rise. Up is positive and down is negative.

4

3. Find the horizontal run. Going right is positive and going left is negative.

74

It does not matter at which point you begin the walk!

x

y1. Write slope ratio.runrise

m

•

•7

Example 5 Find the slope of the line.

2. Start at either point. Find the vertical rise. Up is positive and down is negative.

0

3. Find the horizontal run. Going right is positive and going left is negative.

5

4. Simplify

0

x

y1. Write slope ratio.

runrise

m

• •

zero ooooover the fraction line

is (zeroooooo)

Example 6 Find the slope of the line.

2. Start at either point. Find the vertical rise. Up is positive and down is negative.

4

3. Find the horizontal run. Going right is positive and going left is negative.

0

4. Simplify

undefined

x

y

1. Write slope ratio.

runrise

m

zero unnnnder the fraction line

is unnnnndefine

d

•

•

PRACTICE. Plot the points and draw the line that passes through them. Then find the slope using the slope ratio.1. (2,4) and (2,-4) 3. (3,6) and (3,–1) 5. (6,1) and

(–4,1)2. (4,5) and (2,2) 4. (-2,-3) and (3,-3) 6. (2,2) and

(–1,4)Find the slope by walking the line.

7.•

•

8.

••

x

y

x

yWriting the

slope ratio is a mandatory

step!

PRACTICE. Answers

1. (2,4) and (2,-4) 4. (-2,-3) and (3,-3)2. (4,5) and (2,2) 5. (6,1) and (–4,1)3. (3,6) and (3,–1) 6. (2,2) and (–1,4)

7.•

•

8.

••

x

y

x

y

2 52

undefined 0

23

0

32undefined

4-A2 SKILLS PRACTICE Wkbk. Page 24 #1–19

#4 – 19 Plot the points and draw the line that passes through them. Then find the slope using the slope ratio. Skip #12.