Ch 6 Linear Functions

FeatureLesson

Algebra 1Algebra 1

LessonMain

Chapter 6

Rate of Change and Slope

Algebra IA

Ithaca High School

Ms. Ridgell

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Chapter 6: Rate of Change and SlopeChapter 6: Rate of Change and Slope

Objectives:

1. To find rates of change from tables and graphs.

2. To find slope.

New Vocabulary:

• Rate of change

• Slope

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

Evaluate each function rule for x = –5.

1. y = x – 7 2. y = 7 – x

3. y = 2x + 5 4. y = – x + 3

Write in simplest form.

5. 6. 7.

8. 9. 10.

25

7 – 33 – 1

3 – 56 – 0

8 – (–4) 3 – 7

–1 – 2 0 – 5

–6 – (–4) –2 – 6

0 – 11 – 0

(For help, go to Lessons 5-2 and 2-2.)

Lesson 6-1

Check Skills You’ll Need

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

1. y = x – 7 for x = –5: y = –5 – 7 = –12

2. y = 7 – x for x = –5: y = 7 – (–5) = 12

3. y = 2x + 5 for x = –5: y = 2(–5) + 5 = –10 + 5 = –5

4. y = – x + 3 for x = –5: y = – (–5) + 3 = 2 + 3 = 5

5. = = 2 6. = – = –

7. = – = –3 8. = =

9. = = 10. = – = –1

7 – 33 – 1

3 – 56 – 0

8 – (–4) 3 – 7

–1 – 2 0 – 5

–6 – (–4) –2 – 6

0 – 11 – 0

25

25

42

26

13

12 4

–2–8

14

11

–3–5

35

Lesson 6-1

Solutions

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6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Find the rate of change for each pair of consecutive mileage amounts.

For the data in the table, is the rate of change the

same for each pair of consecutive mileage amounts?

Fee for Miles Driven

Miles Fee

100 $30

150

200

250

$42

$54

$66

Rate of Change and SlopeRate of Change and Slope

Lesson 6-1

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

The rate of change for each pair of consecutive mileage amounts is $12 per 50 miles. The rate of change is the same for all the data.

(continued)

Rate of Change and SlopeRate of Change and Slope

Lesson 6-1

change in cost Cost depends on the change in number of miles number of miles.rate of change =

42 – 30 12 54 – 42 12 66 – 54 12 150–100 50 200–150 50 250–200 50

= = =

Quick Check

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

Below is a graph of the distance traveled by a motorcycle from

its starting point. Find the rate of change. Explain what this rate of

change means.

Choose two pointsOn the graph (0,0) and (400,20)

Lesson 6-1

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

The rate of change is 20 m/s.The motorcycle is traveling 20 meters each second.

vertical change change in distance horizontal change change in time

rate of change =

(continued)

Using the points (0,0) and (400,20), find the rate of change.

400 – 0

20 – 0

= Use two points.

Lesson 6-1

=Divide the vertical change by the horizontal change.

400

20

= Simplify.20

Quick Check

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

The slope of the line is – .32

slope =riserun

Find the slope of each line.

a.

= – 32

= 3 –2

4 – 1

0 – 2

=

Lesson 6-1

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

The slope of the line is 2.

slope =riserun

b. Find the slope of the line.

(continued)

= 2

= –2–1

–1 – 1

–2 – (–2)

=

Lesson 6-1

Quick Check

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

slope =y2 – y1

x2 – x1

The slope of EF is – . 15

Find the slope of each line through E(3, –2) and F(–2, –1).

= – 15

Substitute (–2, –1) for (x2, y2) and(3, –2) for (x1, y1).

–1 – (–2) –2 – 3

=

Lesson 6-1

= 1–5

Simplify.

Quick Check

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

= 0

2 – 2

1 – (–4)

= Substitute (1, 2) for

(x2, y2) and (–4, 2)

for (x1, y1).

slope =y2 – y1

x2 – x1

The slope of the horizontal line is 0.

Find the slope of each line.

a.

Lesson 6-1

= 0 5 Simplify.

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

Division by zero is undefined.

The slope of the vertical line in undefined.

b.

(continued)

Find the slope of the line.

Lesson 6-1

Substitute (2, 1) for (x2, y2) and (2, –4) for (x1, y1).

= 1 – (–4)

2 – 2

= 5 0

Simplify.

slope =y2 – y1

x2 – x1

Quick Check

Additional Examples

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

$15 per ticket400 calories per hour 4

3

75

–

1. Find the rate of change for the data in the table.

5 $75

6

7

8

$90

$105

$120

Tickets Cost

2. Find the rate of change for the data in the graph.

3. Find the slope of the line.

4. Find the slope of the line through (3, –2) and (–2, 5).

Lesson 6-1

Lesson Quiz

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Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

Rate of Change and SlopeRate of Change and Slope

undefined 0

5. State whether the slope is zero or undefined.

a. b.

Lesson 6-1

Lesson Quiz

6-1

Ms. Ridgell2010-11

FeatureLesson

Algebra 1Algebra 1

LessonMain

ReferencesReferences Ms. Ridgell2010-11

•Pearson Prentice Hall Math book, 2008

•Stephen J. Frank, Gull Lake High School, Richland, MI

•David Kapolka, Forest Hills Northern High School, Grand Rapids, MI

•Linda McAleer, Novi High School, Novi, MI

•Matthew McCullen, Big Rapids High School, Big Rapids, MI

•Irene Norde, PhD, Detroit Public Schools, Detroit, MI

•Mary C. O’Connor, Malcom High School, Sault Ste. Marie, MI