Using Tables to Compare Linear and Exponential Growth Functions

5.14.2 Using Tables to Compare Linear and Exponential Growth Functions

5.14.2 • Using Tables to Compare Linear and Exponential Growth Functions

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?

Compare your answer:

The investment grows linearly. It increases by $20 every year.

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

Compare your answer:

$999.60 is 1.02 times $980, and $1,019.59 is 1.02 times $999.60.

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.

Compare your answer:

$b = 1, 000 + 20 x$ where $x$ is the years since the investments were made and $b$ is the value of the bonds.

b. Write an equation for the data in the Savings Account table.

Compare your answer:

$s = 980 \cdot (1.02)^{x}$ where $s$ represents the value of the savings account, in dollars, where and $x$ is the years since the investments were made

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

Compare your answer:

The family should choose bonds if they are investing for less than 12 years. After 12 years, the savings account contains $1,242.88 because this is the value from $980 \cdot (1.02)^{12}$ . And, the bonds are worth $1,240 after 12 years because $1, 000 + 20 (12)$ is the value.

The family should choose the savings account if they are saving for something long term like college or retirement. This is because over time, the savings account will contain more money than the bonds have earned. For instance, after 20 years, the savings account holds $1,456.23 from $[980 \cdot (1.02)^{20}]$ while the bonds only yield $1,400 from $[1, 000 + 20 (20)]$ .

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

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Compare your answer:

A line graph showing years of investment (x-axis) versus amount in dollars (y-axis). Two lines, labeled b (red) and s (blue), both rise steadily, with b increasing slightly faster than s. The two lines intersect at year 11.

Video: Using Tables to Compare Linear and Exponential Functions

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

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Self Check

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

$x$	$(1.02)^x$	$3x$
1	1.02	3
2	1.0404	6
3	1.0612	9
4	1.0824	12
5	1.1041	15

Joanna is incorrect. If you continue the table to $x=300$ , $(1.02)^x$ is greater than $3x$ .
Joanna is incorrect. Since $(1.02)^x$ is exponential growth and $3x$ is linear growth, eventually $(1.02)^x$ will be greater than $3x$ .
Joanna is correct. If you graph the points in the table for $(1.02)^x$ and $3x$ , the points for $3x$ are all above the points for $(1.02)^x$ .
Joanna is correct. None of the values in the table for $(1.02)^x$ are greater than the values for $3x$ .

Additional Resources

Comparing Linear and Exponential Growth Using Tables

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

$x$	$f (x)$
1	$\frac{1}{2}$
2	$\frac{1}{4}$
3	$\frac{1}{8}$
4	$\frac{1}{16}$
5	$\frac{1}{32}$

Table A

$x$	$f (x)$
1	1.4
2	2.5
3	3.6
4	4.7
5	5.8

Table B

$x$	$f (x)$
1	30
2	60
3	120
4	240
5	480

Table C

In each situation, the inputs or $x$ -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 $y$ -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 $y$ -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, $\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}$ . 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 $\frac{1}{2}$ , 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 $2.5 - 1.4 = 1.1$ . 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 $\frac{60}{30} = 2$ . 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.

$x$	$f (x)$
1	2
2	8
3	14
4	20

Table A

$x$	$f (x)$
1	432
2	144
3	48
4	16

Table B

$x$	$f (x)$
1	4
2	16
3	64
4	256

Table C

$x$	$f (x)$
1	6
2	4
3	9
4	2

Table D

Compare your answer:

Here is how to determine which table is exponential growth:

When looking for exponential growth, remember that each $y$ -value should be greater than the previous one. By first checking for growth, you can eliminate Table B and Table D. In Table B, each term is less than the first one, so it is not exponential growth. In Table D, the values increase and decrease, so it is not exponential growth, either.

Next, check for connections between consecutive terms. In Table A, $2 + 6 = 8$ and $2 \cdot 4 = 8$ . Since there seems to be the possibility of either a linear relationship or exponential growth, you need to check the next set of terms. Since $8 \cdot 4 \neq 14$ and $8 + 6 = 14$ , the relationship in Table A is linear, not exponential.

This means the answer should be Table C, but we should always check to confirm. If we check consecutive terms in Table C for a growth factor, we get the following: $\frac{16}{4} = 4$ , $\frac{64}{16} = 4$ , and $\frac{256}{64} = 4$ . This proves that the growth factor is 4 and Table C represents exponential growth.