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## Algebra 1

### Unit 8: Lesson 12

Average rate of change word problems

# Average rate of change review

Review average rate of change and how to apply it to solve problems.

## What is average rate of change?

The average rate of change of function f over the interval a, is less than or equal to, x, is less than or equal to, b is given by this expression:
start fraction, f, left parenthesis, b, right parenthesis, minus, f, left parenthesis, a, right parenthesis, divided by, b, minus, a, end fraction
It is a measure of how much the function changed per unit, on average, over that interval.
It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.

## Finding average rate of change

### Example 1: Average rate of change from graph

Let's find the average rate of change of f over the interval 0, is less than or equal to, x, is less than or equal to, 9:
A coordinate plane. The x- and y-axes each scale by one. The function y equals f of x is a continuous curve that contains the following points: the point negative five, five, the point negative three, zero, the point zero, negative seven, the point two, negative three, the point three, negative three, the point five point five, zero, and the point nine, three. The points zero, negative seven and nine, three are plotted on the function.
We can see from the graph that f, left parenthesis, 0, right parenthesis, equals, minus, 7 and f, left parenthesis, 9, right parenthesis, equals, 3.
\begin{aligned} \text{Average rate of change}&=\dfrac{f(9)-f(0)}{9-0} \\\\ &=\dfrac{3-(-7)}{9} \\\\ &=\dfrac{10}{9} \end{aligned}

### Example 2: Average rate of change from equation

Let's find the rate of change of g, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 9, x over the interval 1, is less than or equal to, x, is less than or equal to, 6.
g, left parenthesis, 1, right parenthesis, equals, 1, cubed, minus, 9, dot, 1, equals, minus, 8
g, left parenthesis, 6, right parenthesis, equals, 6, cubed, minus, 9, dot, 6, equals, 162
\begin{aligned} \text{Average rate of change}&=\dfrac{g(6)-g(1)}{6-1} \\\\ &=\dfrac{162-(-8)}{5} \\\\ &=34 \end{aligned}
Problem 1
What is the average rate of change of g over the interval minus, 8, is less than or equal to, x, is less than or equal to, minus, 2?
A coordinate plane. The x- and y-axes each scale by one. The function y equals g of x is a continuous curve that contains the following points: the point negative eight, negative eight, the point negative five, negative five, the point negative three, zero, the point negative two, three, the point zero, six, the point two, three, the point three, zero, and the point four, negative four. The points negative eight, negative eight and negative two, three are plotted on the function.

Want to try more problems like this? Check out this exercise.