Mini Lesson Question
Compare Linear and Exponential Functions
Look at the patterns in the two tables.
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
| 4 | 16 |
| 5 | 20 |
| 1 | 4 |
| 2 | 8 |
| 3 | 16 |
| 4 | 32 |
| 5 | 64 |
In each situation, the -values are increasing by one. These can also be called consecutive terms.
- If the difference in the -values of consecutive terms is always the same constant, then the values in the table can be modeled by a linear function. The constant is the slope, or rate of change.
- If the quotient for the -values of consecutive terms is always the same constant, then the values in the table can be modeled by an exponential function. The constant is the growth factor.
- It is possible for a table to be a set of values that is neither of these relationships.
Using this information you can see that in the first table, the difference in consecutive -values is the same constant: 8-, , , . The differences are the same, so the table represents a linear function.
In the second table, the quotient of consecutive terms is the same: , . The quotient is the same, so the table represents an exponential function.
Linear growth happens at a constant rate. The rate of change, or slope, in the first table is 4. The linear function is of the form , where m is the slope.
Exponential growth does not happen at a constant rate. The growth factor is constant, but the rate of growth changes as the value of the independent variable, , changes. The exponential function is of the form , where is the growth factor.
Try it
Compare Linear and Exponential Functions.
Look at the patterns in the two tables.
| Table A | |
| 1 | 5 |
| 2 | 15 |
| 3 | 45 |
| 4 | 135 |
| 5 | 405 |
| Table B | |
| 1 | 5 |
| 2 | 15 |
| 3 | 25 |
| 4 | 35 |
| 5 | 45 |
Which of the following tables represents linear growth and which represents exponential growth? Be prepared to show your reasoning.
Here is how to determine which table represents linear growth and which represents exponential growth:
Check for connections between consecutive terms.
- In Table A, and . There seems to be the possibility of either a linear relationship or exponential growth, so you need to check the next set of terms. but , so the relationship in Table A is exponential. To be sure, check the rest of the values in the table.
- n Table B, and . There seems to be the possibility of either a linear relationship or exponential growth, so you need to check the next set of terms. but , so the relationship in Table B is linear. To be sure, check the rest of the values in the table.
Check Your Understanding
Which statement best describes the following tables?
| 1 | 2 |
| 2 | 6 |
| 3 | 18 |
| 4 | 54 |
| 5 | 162 |
| 1 | 2 |
| 2 | 6 |
| 3 | 10 |
| 4 | 14 |
| 5 | 18 |
Table A represents an exponential function with a constant growth factor of 3; Table B represents a linear function with a constant difference of 4.
Table A represents a linear function with a constant difference of 4; Table B represents an exponential function with a constant growth factor of 3.
Table A represents a linear function with a constant growth factor of 3; Table B represents an exponential function with a constant difference of 4.
Table A represents an exponential function with a constant difference of 4; Table B represents a linear function with a constant growth factor of 3.
Take a moment to think about what you learned in the mini-lesson review.
Video: Linear and Exponential Growth
Watch the following video to learn more about linear and exponential growth.
Khan Academy: Exponential VS Linear Growth