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Algebra 1

Compare Linear and Exponential Functions: Mini-Lesson Review

Algebra 1Compare Linear and Exponential Functions: Mini-Lesson Review

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Mini Lesson Question

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

 

  1. 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.
  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.
  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.
  4. 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 x y y
1 4
2 8
3 12
4 16
5 20
x x y y
1 4
2 8
3 16
4 32
5 64

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

  • If the difference in the y 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 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 y -values is the same constant: 8- 4 = 4 4 = 4 , 12 8 = 4 12 8 = 4 , 16 12 = 4 16 12 = 4 , 20 16 = 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: 8 4 = 2 8 4 = 2 , 16 8 = 2 , 32 16 = 2 , 64 32 = 2 16 8 = 2 , 32 16 = 2 , 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 y = m x + b , where m is the slope.

Exponential growth that 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 x , changes. The exponential function is of the form y = a b x y = a b x , where b b is the growth factor.

Try it

Try It: Compare Linear and Exponential Functions.

Look at the patterns in the two tables.

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

Check Your Understanding

Which statement best describes the following tables?

x x y y
1 2
2 6
3 18
4 54
5 162
Table A
x x y y
1 2
2 6
3 10
4 14
5 18
Table B

Multiple Choice:

  1. 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.

  2. 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.

  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.

  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.

Video: Linear and Exponential Growth

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

Khan Academy: Exponential VS Linear Growth

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