Activity
The graphs show the populations of California and Texas over time.
1. Estimate the average rate of change in the population in California between 1970 and 2010. Be prepared to show your reasoning.
Compare your answer:
The population of California was about 20 million in 1970 and 37 million in 2010. The average rate of change is: million per year.
2. In this situation, what does the rate of change mean for California?
Compare your answer:
On average, the population of California was growing by 425,000 people per year.
3. Estimate the average rate of change in the population in Texas between 1970 and 2010. Be prepared to show your reasoning.
Compare your answer:
The population of Texas was about 11 million in 1970 and 25 million in 2010. The average rate of change is million per year.
4. In this situation, what does the rate of change mean for Texas?
Compare your answer:
On average, the population of Texas was growing by 350,000 people per year.
5. Which state's population grew more quickly between 1900 and 2000? Be prepared to show your reasoning.
Compare your answer:
California is growing more quickly. For example: If we connect each pair of points representing the population of each state in 1900 and 2000, the segment for California has a greater slope, so it has a greater average rate of change.
Video: Interpreting Average Rate of Change
Watch the following video to learn more about interpreting the average rate of change.
Self Check
Additional Resources
Interpreting Slope
A Mars rover collected the following temperature data over 1.6 Martian days. A Martian day is called a Sol. Use the graph to answer the following questions.
Which had a greater average rate of change between 10.2 and 10.4 Sol, ground temperature or air temperature? What is the meaning of each rate change?
Since ground temperature has a greater slant or slope, it has a greater rate of change. Slope is another word for the rate of change of a graph.
To find the average rate of change, first you need to find Sol on the graph (which is what a day is called on Mars). Find the values of 10.2 and 10.4 on the -axis. Find the corresponding -values of the points for both the ground temperature and the air temperature graphs. By finding the average rate of change, you will be able to determine which one was greater.
Ground Temperature:
Air Temperature:
Substitute each set of points into the average rate of change formula:
Ground Temperature =
Air Temperature =
Remember that to interpret the meaning of the rate of change, you need to compare the axes. It is the change in the -axis units divided by the change in the -axis units, or . That means that the average rate of change of each can be interpreted as:
Ground Temperature’s average rate of change is 350 degrees celsius per Sol (or day on Mars).
Air Temperature’s average rate of change is 150 degrees celsius per Sol (or day on Mars).
Try it
Try It: Interpreting Slope
A Mars rover collected the following temperature data over 1.6 Martian days. A Martian day is called a Sol. Use the graph to answer the following questions.
Which had a greater average rate of change between 10.6 and 11 Sol, ground temperature or air temperature? What is the meaning of each rate change?
Here is how to find the average rate of change:
To find the average rate of change, first you need to find Sol on the graph (which is what a day is called on Mars). Find the values of 10.6 and 11 on the -axis. Find the corresponding -values of the points for both the ground temperature and air temperature graphs. By finding the average rate of change, you will be able to determine which one was greater.
Ground Temperature:
Air Temperature:
Substitute each set of points into the average rate of change formula:
Ground Temperature =
Air Temperature =
Remember that to interpret the meaning of the rate of change, you need to compare the axes. It is the change in the -axis units divided by the change in the -axis units, or . That means that each average rate of change of each can be interpreted as:
Ground temperature’s average rate of change is –205 degrees celsius per Sol (or day on Mars).
Air temperature’s average rate of change is –150 degrees celsius per Sol (or day on Mars).
The ground temperature had a greater rate of decrease per Sol.