Topics
3. Linear Functions and Equations
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United States of America
Grades 9-12

3.03 Average rate of change

Lesson
Practice

Introduction

In 8th grade, we used the slope of a linear function to describe its rate of change and we will connect that to the average rate of change in a non-linear function. By calculating the slope over a given interval, we can find the average rate of change of the function on that interval. An understanding of units will help us interpret the meaning of the average rate of change in a variety of contexts.

Average rate of change

To get an idea of how the graph of the function changes, we can take the average rate of change over a specific interval of the domain.

Average rate of change

The change in the value of the dependent variable per unit change in the independent variable over a given interval

The average rate of change of a function can be calculated by dividing the change in function values between the start and the end of the interval by the length of the interval.

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x
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f(x)

For the function shown, the function values increase from 3 to 7 over the interval 1 \leq x \leq 5.

This means that over the interval 1 \leq x \leq 5, the function has an average rate of change of \dfrac{7 - 3}{5 - 1} = 1

Note that the endpoints of the interval are (1,3) and (5,7).

On the graph, the dashed lines show an increase of 1 unit in the y-values per unit increase in the x-values over this interval.

Suppose the values of the given function from the graph were represented in the table shown:

x0135
f(x)0.535.87

In order to find the average rate of change over the interval 1 \leq x \leq 5 in the table, we could again calculate \dfrac{7-3}{5-1}=1

Recall that the slope of a linear function is also its rate of change. For a linear function, the slope is always the same, therefore we say that the rate of change is constant.

For a non-linear function, the rate of change varies along with the graph. Therefore, we say a non-linear function has a variable rate of change.

To find the average rate of change from a given function over the interval a \leq x \leq b, we can find the change in the value of the dependent variable f(b)-f(a) per change in value in the independent variable b-a, or:

\dfrac{f(b)-f(a)}{b-a}

The rate of change in a graph can be positive or negative. The lines below have positive rates of change. Notice how as the values on the x-axis increase, the values on the y-axis also increase.

Three four quadrant coordinate planes. In the left coordinate plane, a line is passing the origin and leaning to the right. In the middle coordinate plane, a line is passing (0, -1) and leaning to the right but steeper than the line in the left. In the right coordinate plane, a line is passing (0, 4) and leaning to the right but less steep than the line in the left.

These next graphs have negative rates of change. Unlike graphs with a positive slope, as the values on the x-axis increase, the values on the y-axis decrease.

Three four quadrant coordinate planes. In the left coordinate plane, a line is passing the origin and leaning to the left. In the middle coordinate plane, a line is passing (0, 4) and leaning to the right but steeper than the line in the left. In the right coordinate plane, a line is passing (0, negative 2) and leaning to the right but less steep than the line in the left.

Non-linear functions may have intervals where they are increasing or decreasing.

Examples

Example 1

A flock of birds migrate to a new island. The population of birds on that island over the next six years is shown on the graph.

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\text{Birds}

Calculate and interpret the average rate of change of the bird population over the six-year period.

Worked Solution
Create a strategy

To find the average rate of change, we want to divide the difference between the initial and final populations by the length of the period.

Apply the idea

Looking at the graph we can see that the initial population is 15 birds, and after 6 years the population is 60 birds.

So we can calculate the average rate of change as \frac{60 - 15 \text{ birds}}{6 \text{ years}} = \frac{7.5 \text{ birds}}{1 \text{ year}}

Therefore, the average rate of change is 7.5 birds per year, meaning that on average, we can expect that the population grew by about 7.5 birds yearly.

Reflect and check

Don't forget to include units when working with a real-world context, since the average rate of change is still a rate.

Example 2

Consider the function f\left(x\right) = 2x^2 - 1.

Calculate the average rate of change over the interval -4 \leq x \leq 0.

Worked Solution
Create a strategy

The average rate of change can be thought of as:\dfrac{\text{Change in }f\left(x\right)}{\text{Change in }x}

To determine the change in f\left(x\right), we want to identify the values of f\left(x\right) when x=-4 and when x=0.

Apply the idea

Evaluate at x=-4

\displaystyle f\left(x\right)\displaystyle =\displaystyle 2\left(x\right)^2 - 1Original equation
\displaystyle f\left(-4\right)\displaystyle =\displaystyle 2\left(-4\right)^2 - 1Sustitute x=-4
\displaystyle =\displaystyle 2 \cdot 16 - 1Evaluate the exponent
\displaystyle =\displaystyle 31Evaluate the multiplication and subtraction

Evaluate at x=0

\displaystyle f\left(x\right)\displaystyle =\displaystyle 2\left(x\right)^2 - 1Original equation
\displaystyle f\left(0\right)\displaystyle =\displaystyle 2\left(0\right)^2 - 1Substitute x=0
\displaystyle =\displaystyle -1Evaluate

Going back to the idea that the average rate of change can be thought of as \dfrac{\text{Change in }f\left(x\right)}{\text{Change in }x}, we have:

\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \frac{\text{Change in }f\left(x\right)}{\text{Change in }x}Average rate of change formula
\displaystyle =\displaystyle \dfrac{-1 - 31}{0 -(-4)}Substitute values of f\left(x\right) and x from above
\displaystyle =\displaystyle \dfrac{-32}{4}Evaluate the subtraction
\displaystyle =\displaystyle -8Evaluate the divison

Example 3

Shown in the table below is the vertical distance of a baseball that is thrown from the height of a player.

Time (seconds)00.250.500.7511.251.501.75
Height (feet)613182122211813
a

Calculate and interpret the average rate of change from 0 seconds to 1 second.

Worked Solution
Create a strategy

To find the average rate of change we can use the endpoints of the interval with the average rate of change formula:

\dfrac{f(b)-f(a)}{b-a}The endpoints of the interval are at f(0) and f(1).

Apply the idea

We can use the table to find that f(0)=6 and f(1)=22 and substitute these values into the formula.

\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \dfrac{f(b)-f(a)}{b-a}Average rate of change formula
\displaystyle =\displaystyle \dfrac{f(1)-f(0)}{1-0}Substitute b=1 and a=0
\displaystyle =\displaystyle \dfrac{22-6}{1-0}Substitute f\left(1\right)=22 and f\left(0\right)=6
\displaystyle =\displaystyle \dfrac{16}{1}Evaluate the subtraction
\displaystyle =\displaystyle 16Evaluate the division

Therefore, the average rate of change of the ball is 16 feet per second from 0 to 1 second. This means that the ball's vertical distance was increasing at this time.

Reflect and check

A graph of the function can help us see the vertical position of the baseball, and we might estimate the average rate of change of the interval by using the difference in the values of the function along with the interval 0 \leq x \leq 1.

Height of a Baseball
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Find and compare the rate of change of the ball from 1 second to 1.25 seconds and from 1.25 seconds to 1.75 seconds.

Worked Solution
Create a strategy

To compare the rates of change we will need to find the rate of change for each interval separately.

The endpoints of the first interval are at f(1) and f(1.25). The endpoints of the second interval are at f(1.25) and f(1.75).

Apply the idea

We can use the information from the table to evaluate the average rate of change for the interval. Based on the table, f(1)=22 and f(1.25)=21. So,\dfrac{f(b)-f(a)}{b-a}=\dfrac{f(1.25)-f(1)}{1.25-1}=\dfrac{21-22 \text{ feet}}{1.25-1 \text{ seconds}}=\dfrac{-1 \text{ feet}}{0.25 \text{ seconds}}=\dfrac{-4 \text{ feet}}{1 \text{ second}}Based on the table, f(1.25)=21 and f(1.75)=13. So,\dfrac{f(b)-f(a)}{b-a}=\dfrac{f(1.75)-f(1.25)}{1.75-1.25}=\dfrac{13-21 \text{ feet}}{1.75-1.25 \text{ seconds}}=\dfrac{-8 \text{ feet}}{0.50 \text{ seconds}}=\dfrac{-16 \text{ feet}}{1 \text{ second}}Therefore, the average rate of change of the ball is -4 feet per second from 1 to 1.25 seconds and -16 feet per second from 1.25 to 1.75 seconds. This means that the ball's vertical distance was decreasing during both time intervals, but its average rate of change was faster from 1.25 to 1.75 seconds.

Reflect and check

A graph of the function can give us a visual of the ball's vertical distance within the given time intervals. Based on the outlined intervals, the graph is steeper in the second interval, where we calculated that the average rate of change was indeed faster.

Height of a Baseball
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\text{Seconds}
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\text{Feet}
Idea summary

The average rate of change can be calculated from a graph, equation, or table. To calculate the average rate of change we divide the total change in the function values by the length of the interval:

\dfrac{f(b)-f(a)}{b-a}

We should use units to help guide our interpretation of the average rate of change in context.

Outcomes

F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

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