Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

12
Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change

Transcript of Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Page 1: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Lesson 3.4

Constant Rate of Change

(linear functions)

1

3.3.2: Proving Average Rate of Change

Page 2: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Introduction

• A rate of change is a ratio that describes how much

one quantity changes with respect to the change in

another quantity of the function.

• With linear functions the rate of change is called the

slope. The slope of a line is the ratio of the change in

y-values to the change in x-values. Formula: m =

• Linear functions have a constant rate of change,

meaning values increase or decrease at the same

rate over a period of time. 2

3.3.2: Proving Average Rate of Change

Page 3: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Recall…..

• The rate of change between any two points of a linear function will be equal.

3

3.3.2: Proving Average Rate of Change

CalculatingConstant Rate of Change (slope)

from a Table1. Choose two points from the table.

2. Assign one point to be (x1, y1) and the other point to be (x2, y2).

3. Substitute the values into the slope formula:

4. The result is the rate of change for the interval between the two points chosen.

Page 4: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice

Example 1To raise money, students

plan to hold a car wash.

They ask some adults how

much they would pay for

a car wash. The table on

the right shows the results

of their research. What is the

rate of change for their results?

4

3.3.2: Proving Average Rate of Change

Carwash Price (x)

Number of Customers (f(x))

$4 120

$6 105

$8 90

$10 75

Page 5: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice: Example 1, continued1. Choose two points from the table.

(4, 120) and (10, 75)

2. Assign one point to be (x1, y1) and the other

to be (x2, y2). It doesn’t matter which is which. Let (4, 120) be (x1, y1) and

(10,76) be (x2, y2). 5

3.3.2: Proving Average Rate of Change

Carwash Price (x)

Number of Customers (f(x))

$4 120

$6 106

$8 92

$10 78

Page 6: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice: Example 1, continued

3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change.

Slope formula

Substitute (4, 120) and (10, 78)

for (x1, y1) and (x2, y2).

Simplify as needed.

6

3.3.3: Recognizing Average Rate of Change

¿78−12010−4

¿−426

= -7 The rate of change for this function is -7 customers per dollar. For every dollar the carwash price increases, 7 customers are lost.

Page 7: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Recall….

• The rate of change between any two points of a linear function will be equal.

7

3.3.2: Proving Average Rate of Change

Estimating Constant Rate of Change (slope)

from a Graph1. Pick two points from the graph.

2. Identify (x1, y1) as one point and (x2, y2) as the other point.

3. Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change.

4. The result is the estimated constant rate of change (slope) for the graph.

m =

Page 8: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

You Try

Calculate the constant rate of change (slope) for these tables.

8

3.3.2: Proving Average Rate of Change

x f(x)

1 -6

2 -11

3 -16

4 -21

x f(x)

-3 -5

0 -4

3 -3

6 -2

1) 2)

Page 9: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice

Example 2The graph to the right

compares the distance a

small motor scooter can

travel in miles to the amount

of fuel used in gallons.

What is the rate of change

for this scenario?

9

3.3.3: Recognizing Average Rate of Change

Page 10: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice: Example 2, continued

1. Pick two points from the graph. The function is linear, so the rate of change will be constant for any interval (continuous portion) of the function.

Choose points on the graph with coordinates that are easy to estimate. For example, (0, 1.5) and (155,0)

2. Identify (x1, y1) as one point and (x2, y2) as the other point. It doesn’t matter which is which.

Let’s have (0, 1.5) be (x1, y1) and (155,0) be (x2, y2) 10

3.3.3: Recognizing Average Rate of Change

Page 11: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

Guided Practice: Example 2, continued

3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change.

Slope formula

Substitute (0,1.5) and (155, 0)

for (x1, y1) and (x2, y2).

Simplify as needed.

11

3.3.3: Recognizing Average Rate of Change

¿0−1.5155−0

¿−1.5155

≈ -0.01

Page 12: Lesson 3.4 Constant Rate of Change (linear functions) 1 3.3.2: Proving Average Rate of Change.

You Try

Calculate the constant rate of change (slope) for these graphs.

12

3.3.2: Proving Average Rate of Change

1) 2)