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Algebra 1

5.14.2 Using Tables to Compare Linear and Exponential Growth Functions

Algebra 15.14.2 Using Tables to Compare Linear and Exponential Growth Functions

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Activity

A family has $1,000 to invest and is considering two options: investing in government bonds that offer 2% simple interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound interest. Both options pay interest annually.

Here are two tables showing what they would earn in the first couple of years if they do not invest additional amounts or withdraw any money.

years of investment amount in dollars
0 $1,000
1 $1,020
2 $1,040
Bonds
years of investment amount in dollars
0 $980
1 $999.60
2 $1,019.59
Savings Account

1. Bonds: How does the investment grow with simple interest?

2. Savings account: How are the amounts $999.60 and $1,019.59 calculated?

3. Write an equation to represent the relationship between the amount of money and the number of years of investment for each option.

a. Write an equation for the data in the Bonds table. Use the “^” symbol to enter an exponent, if needed.

b. Write an equation for the data in the Savings Account table. Use the “^” symbol to enter an exponent, if needed.

4. Which investment option should the family choose? Use your equations or calculations to support your answer.

5. Use the Desmos graphing tool to graph the two investment options and show how the money grows in each.

Video: Using Tables to Compare Linear and Exponential Functions

Watch the following video to learn more about comparing linear and exponential functions:

Self Check

Using a calculator, Joanna made the following table and then made the following conjecture: 3 x is always greater than ( 1.02 ) x . Is Joanna correct? Explain.

x ( 1.02 ) x 3 x
1 1.02 3
2 1.0404 6
3 1.0612 9
4 1.0824 12
5 1.1041 15
  1. Joanna is incorrect. If you continue the table to x = 300 , ( 1.02 ) x is greater than 3 x .
  2. Joanna is incorrect. Since ( 1.02 ) x is exponential growth and 3 x is linear growth, eventually ( 1.02 ) x will be greater than 3 x .
  3. Joanna is correct. If you graph the points in the table for ( 1.02 ) x and 3 x , the points for 3 x are all above the points for ( 1.02 ) x .
  4. Joanna is correct. None of the values in the table for ( 1.02 ) x are greater than the values for 3 x .

Additional Resources

Comparing Linear and Exponential Growth Using Tables

Classify each of the following tables as linear, exponential growth, or exponential decay.

xx f(x)f(x)
1 1212
2 1414
3 1818
4 116116
5 132132
Table A
xx f(x)f(x)
1 1.4
2 2.5
3 3.6
4 4.7
5 5.8
Table B
xx f(x)f(x)
1 30
2 60
3 120
4 240
5 480
Table C

In each situation, the inputs or xx-values are increasing by 1 each time. These can also be called consecutive terms.

If you have consecutive terms, then:

  • If the difference between their corresponding outputs or yy-values is always the same constant, then the input-output pairs in the table can be modeled by a linear function.
  • If the quotient between their corresponding outputs or yy-values is always the same constant, then the input-output pairs in the table can be modeled by an exponential function. A growth rate greater than one means that it is exponential growth, and a growth rate between zero and one means that it is exponential decay.

It is possible for a table to be a set of values that is not any of these relationships.

Using this information, you can see that in Table A, 1412=121412=12. This holds true for the quotient of every other set of consecutive terms, too. Since the quotient is always the same, Table A represents an exponential function. Since the growth rate is 1212, or between zero and one, Table A represents exponential decay.

In Table B, the quotient of consecutive terms is not the same, but there is a common difference. The difference between the first two terms is 2.51.4=1.12.51.4=1.1. This holds true for each of the other pairs of consecutive terms, so Table B represents a linear relationship.

In Table C, the quotient of consecutive terms is 6030=26030=2. This relationship holds true for all of the pairs of consecutive terms, which means this table represents an exponential relationship. Since the growth factor is greater than one, Table C represents exponential growth.

Try it

Try It: Comparing Linear and Exponential Growth Using Tables

Which of the following tables represents exponential growth? Be prepared to show your reasoning.

xx f(x)f(x)
1 2
2 8
3 14
4 20
Table A
xx f(x)f(x)
1 432
2 144
3 48
4 16
Table B
xx f(x)f(x)
1 4
2 16
3 64
4 256
Table C
xx f(x)f(x)
1 6
2 4
3 9
4 2
Table D

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