Lesson 3.4Constant Rate of Change

(linear functions)

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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.

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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.

Guided PracticeExample 1To raise money, studentsplan to hold a car wash.They ask some adults howmuch they would pay for a car wash. The table on the right shows the resultsof their research. What is therate of change for their results?

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3.3.2: Proving Average Rate of Change

Carwash Price (x)

Number of Customers (f(x))

$4 120$6 105$8 90

$10 75

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 otherto 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

Guided Practice: Example 1, continued3. 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.

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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.

Recall….

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

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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 =

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

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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)

Guided PracticeExample 2The graph to the rightcompares the distance asmall motor scooter cantravel in miles to the amountof fuel used in gallons. What is the rate of change for this scenario?

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3.3.3: Recognizing Average Rate of Change

Guided Practice: Example 2, continued1. 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)

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3.3.3: Recognizing Average Rate of Change

Guided Practice: Example 2, continued3. 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.

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3.3.3: Recognizing Average Rate of Change

¿0−1.5155−0

¿−1.5155

≈ -0.01

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

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3.3.2: Proving Average Rate of Change

1) 2)