# Average Rate of Change Lesson

Average rate of change, is a mathematical concept that helps us understand how a quantity is changing over a specific time interval. It allows us to calculate the average rate at which something is changing, such as the speed of an object or the rate of growth of a population. To calculate the average rate of change, we need to know the initial and final values of the quantity we are studying. We then divide the change in the quantity by the change in time.

For example, if we wanted to find the average rate of change of the distance traveled by a car over a 3-hour period, we would divide the change in distance by 3. The average rate of change can also be represented graphically. If we have a graph of the quantity over time, the average rate of change represents the slope of the line connecting the initial and final points on the graph.

Average rate of change can also be represented using a table. In a table, the average rate of change is determined by the change in the dependent variable divided by the independent variable. For example, if we had a table that showed the amount of money made over a period of time, we would divide the change in money made divided by the number of hours worked.

The average rate of change is calculated by finding the change in the y-values divided by the change in the x-values. Symbolically, this is denoted as A(x) = Δ y / Δ x. Change means the difference, so in this case it is the difference in the y-values divided by the difference in x-values. As a reminder, difference means to subtract.

Some other examples of average rate of change are:

• Sally baked three dozen cookies per hour.
• Charlie drove 45 miles per hour.
• The car got 20 miles per gallon of gas.

Example 1: Find the average rate of change given the following graph.

Solution:

The initial point is at the origin, (0, 0), and the final point is at (4, 24). So the change in y-values from the initial point is 24 miles:

△y = 24 – 0 = 24

and the change in x-values from the initial point is 4 hours

△x = 4 – 0 = 4

Therefore, A(x) = △y/△x = 24/4 = 6

The average rate of change is 6 miles per hour.

Example 2: Find the average rate of change given the following table.

Solution:

The initial point is (20, 1) and the final point is (60, 3). Therefore, we can find the change in y-values and the change in x-values, as seen below.

△y = 3 – 1 = 2

△x = 60 – 20 = 40

Therefore, A(x) = △y/△x = 40/2 = 20.

The average rate of change is 20 miles per gallon of gas.

Example 3: Find the average rate of change if a person starts baking cookies, and bakes 5 dozen in the first hour, and bakes a total of 20 dozen cookies after 4 hours.

Solution:

The initial point would be (5, 1), and the final point would be (20, 4). Therefore, we can find the change in y-values and the change in x-values, as seen below.

△y = 4 – 1 = 3

△x = 20 – 5 = 15

Therefore, A(x) = △y/△x = 15/3 = 5

The average rate of change is 5 dozen cookies per hour.

1. How is the average rate of change calculated?

The average rate of change is calculated by finding the difference in the y-values over a given interval and dividing that difference by the corresponding change in the x-values.

1. Can average rate of change be negative?

Yes, the average rate of change can be negative. It simply indicates that the quantity being measured is decreasing over the given interval.

1. What does a positive/negative average rate of change mean?

A positive average rate of change indicates an increase in the quantity being measured over a given interval, while a negative average rate of change implies a decrease.

1. Is there a formula for calculating average rate of change?

Yes, the formula for average rate of change is often expressed as: