Activity
Here are two functions: and .
Investigate the output of and for different values of . For large enough values of , one function will have a greater value than the other.
Which function will have a greater value as increases?
Support your answer with tables, graphs, or other representations.
Compare your answer:
is larger when and again when . Once is larger than , is growing much more quickly, as shown in the table.
Alternatively, here is a graph showing the plotted values from the table and another showing and .
Are you ready for more?
Extending Your Thinking
Janna says that some exponential functions grow more slowly than the quadratic function as increases. Do you agree with Janna? Be prepared to show your reasoning.
Compare your answer:
Yes. This is true for the functions and as increases for -values less than .
Could you have an exponential function so that for all values of ?
Compare your answer:
No, for all values of (except ), because and . (If , there are problems if , so this is not usually considered an exponential function.)
Video: Comparing Exponential and Quadratic Functions
Watch the following video to learn more about functions.
Self Check
Additional Resources
Exponential versus Quadratic Functions
For each of the two functions, do the following:
- Complete the table of values.
- Sketch a graph.
- Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know.
or equivalent |
This function is exponential. It has a growth factor of .
This function is not linear because it does not have a constant rate of change, and it is not exponential because it does not have a constant growth factor. The function is quadratic because it can be represented by , which has to the second power, and that is the greatest power.
Note: Between and , will have the greater value as increases.
Try it
Try It: Exponential versus Quadratic Functions
Which of the following has the greater value as increases, or ?
Here is how to determine which function has the greater value as increases:
Make a table of values for each function, and then graph the function:
Looking at both the table of values and the graphs, will have a greater value than as increases. This is because is exponential and has a constant growth factor.