Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Compare Linear and Exponential Functions: Mini-Lesson Review

Mini Lesson Question

Question #3: Compare Linear and Exponential Functions

Which statement best describes the following tables?

$x$	$y$
1	3
2	6
3	9
4	12
5	15

Table A

$x$	$y$
1	3
2	6
3	12
4	24
5	48

Table B

Table A represents an exponential function with a constant difference of 3; Table B represents a linear function with a constant growth factor of 2.
Table A represents a linear function with a constant growth factor of 2; Table B represents an exponential function with a constant difference of 3.
Table A represents an exponential function with a constant growth factor of 2; Table B represents a linear function with a constant difference of 3.
Table A represents a linear function with a constant difference of 3; Table B represents an exponential function with a constant growth factor of 2.

Compare Linear and Exponential Functions

Look at the patterns in the two tables.

$x$	$y$
1	4
2	8
3	12
4	16
5	20

$x$	$y$
1	4
2	8
3	16
4	32
5	64

In each situation, the $x$ -values are increasing by one. These can also be called consecutive terms.

If the difference in the $y$ -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 $y$ -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 $y$ -values is the same constant: 8- $4 = 4$ , $12 - 8 = 4$ , $16 - 12 = 4$ , $20 - 16 = 4$ . The differences are the same, so the table represents a linear function.

In the second table, the quotient of consecutive terms is the same: $\frac{8}{4} = 2$ , $\frac{16}{8} = 2, \frac{32}{16} = 2, \frac{64}{32} = 2$ . 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 $y = m x + b$ , 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, $x$ , changes. The exponential function is of the form $y = a b^{x}$ , where $b$ is the growth factor.

Try it

Compare Linear and Exponential Functions.

Look at the patterns in the two tables.

Table A
$x$	$y$
1	5
2	15
3	45
4	135
5	405

Table B
$x$	$y$
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, $5 + 10 = 15$ and $5 \cdot 3 = 15$ . There seems to be the possibility of either a linear relationship or exponential growth, so you need to check the next set of terms. $15 + 10 \neq 45$ but $15 \cdot 3 = 45$ , so the relationship in Table A is exponential. To be sure, check the rest of the values in the table.

n Table B, $5 + 10 = 15$ and $5 \cdot 3 = 15$ . There seems to be the possibility of either a linear relationship or exponential growth, so you need to check the next set of terms. $15 + 10 = 25$ but $15 \cdot 3 \neq 25$ , 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?

$x$	$y$
1	2
2	6
3	18
4	54
5	162

Table A

$x$	$y$
1	2
2	6
3	10
4	14
5	18

Table B

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.

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.

Check yourself: Check for connections between consecutive terms.

In Table A, the $x$ -values consistently increase by 1 unit. The difference between the $y$ -values varies: between 6 and 2 is 4 units, between 18 and 6 is 12 units, etc. Since there is no common difference, Table A is not linear. But when the ratios of the consecutive terms are compared: $\frac{6}{2}$ , $\frac{18}{6}$ , $\frac{54}{18}$ , $\frac{162}{54}$ , etc. they all simplify to 3. This could also be identified by noticing that 2 times 3 yields 6; and, 6 times 3 yields 18; and 18 times 3 yields 54, etc. This means Table A is an exponential function with a growth factor of 3.
In Table B, again, the $x$ -values consistently increase by 1 unit. Now, when the difference between the $y$ -values is compared, they are equivalent: $6 - 2 = 4$ ; $10 - 6 = 4$ ; $18 - 14 = 4$ ; etc. Thus, Table B is a linear function with a common difference of 4.

Video: Linear and Exponential Growth

Watch the following video to learn more about linear and exponential growth.

Khan Academy: Exponential VS Linear Growth

Examples of linear and exponential relationships