Compare Linear Functions with Exponential Functions

5.14.3 Compare Linear Functions with Exponential Functions

5.14.3 • Compare Linear Functions with Exponential Functions

Activity

Consider the functions $f (x) = 2 x$ and $g (x) = (1.01)^{x}$ .

1. Complete the table of values for the functions $f$ and $g$ .

$x$	$f (x)$	$g (x)$
1
10
50
100
500

Compare your answer:

$x$	$f (x)$	$g (x)$
1	2	1.01
10	20	~1.10
50	100	~1.645
100	200	~2.7
500	1,000	~144.8

2. Based on the table of values, which function do you think grows faster? Be prepared to show your reasoning.

Compare your answer:

$f$ is growing faster for the values of $x$ in the table, but as $x$ increases, $g$ starts to grow faster. So as $x$ increases, $g$ may eventually grow faster than $f$ .

3. Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing $x$ by 100 a few times and record the function values in the table.

Compare your answer:

When $x$ is 600, $f (600) = 1, 200$ and $g (600) \approx 392$ , so $f$ takes a larger value. When $x$ is 700, $f (700) = 1, 400$ and $g (700) \approx 1, 059$ , so $f$ still takes a larger value. But when $x$ is 800, $f (800) = 1, 600$ and $g (800) \approx 2, 865$ , so $g$ takes a larger value.

Are you ready for more?

Extending Your Thinking

Consider the functions $g (x) = x^{5}$ and $f (x) = 5^{x}$ . While it is true that $f (7) > g (7)$ , for example, it is hard to check this using mental math. Find a value of $x$ for which properties of exponents allow you to conclude that $f (x) > g (x)$ without a calculator.

Compare your answer:

Your answer may vary, but here are some samples: One example is $x = 25$ . We have $f (25) = 5^{25}$ , whereas $g (25) = 25^{5} = (5^{2})^{5} = 5^{10}$ , which is certainly smaller than $5^{25}$ .

Self Check

Terence looked down the second column of the table below and noticed that $\frac{3}{1}=\frac{9}{3}=\frac{27}{9}=\frac{81}{27}$ . Because of his observation, he claimed that the input-output pairs in this table could be modeled with an exponential function. Explain why Terence is correct or incorrect.

$x$	$T(x)$
0	1
1	3
4	9
13	27
40	81

Terence is incorrect. This table represents a linear relationship since the ratio of change of each row is consistent.
Terence is incorrect. This table represents points that do not have an exponential or linear relationship because the $x$ and $y$ change by different amounts.
Terence is correct. This table can be modeled by $y=1(3)^x$ .
Terence is correct. Terms in a table that are connected by a factor are exponential.

Additional Resources

Determining and Comparing Linear and Exponential Functions

Mr. Smith has an apple orchard. He hires his daughter, Lucy, to pick apples and offers her two payment options:

Option A: $1.50 per bushel of apples picked

Option B: 1 cent for coming to work, 3 cents for picking one bushel, 9 cents for picking two bushels, 27 cents for picking three bushels, and so on, with the amount of money tripling for each additional bushel picked

Which option is linear, and which option is exponential? How do you know?
Create a table to model each scenario.
If she picks 6 bushels, which option is better?
If she picks 12 bushels, which option is better?
How many bushels does she need to pick for Option B to be better than Option A?

Let’s first think about linear and exponential functions in general. Let’s also consider Options A and B when $x$ represents the bushels of apples picked and $f (x)$ represents the total amount she earns based on $x$ bushels of apples.

Since Option A is $1.50 per bushel picked, it is linear. For Option B, the amount of money is tripling for each bushel picked, so it is exponential.

	Linear Model	Exponential Model
General Form	$f (x) = a x + b$	$f (x) = a (b)^{x}$
Meaning of Parameters $a$ and $b$	$a$ is the slope of the line or the constant rate of change; $b$ is the $y$ -intercept or the $f (x)$ value at $x = 0$ .	$a$ is the $y$ -intercept or the $f (x)$ value when $x = 0$ ; $b$ is the base or the constant quotient of change.
Example	$f (x) = 1.50 x$	$f (x) = .01 (3)^{x}$
Rule for Finding $f (x + 1)$ from $f (x)$	Starting at $(0, 0)$ , to find $f (x + 1)$ , add 1.5 to $f (x)$ .	Starting at $(0, 0.1)$ , to find $f (x + 1)$ , multiply $f (x)$ by 3.

$x$	$f (x)$
0	0
1	1.50
2	3.00
3	4.50

Linear Model Table

$x$	$f (x)$
0	0.01
1	0.03
2	0.09
3	0.27

Exponential Model Table

To create the table for Option A, add $1.50 to each consecutive term. To create the table for Option B, multiply each term by 3 to get the next term.

Option A table

Number of bushels	Amount of money earned
1	1.50
2	3.00
3	4.50
4	6.00
5	7.50
6	9.00

Option B table

Number of bushels	Amount of money earned
1	0.03
2	0.09
3	0.27
4	0.81
5	2.43
6	7.29

To determine which scenario is better when she picks 6 bushels, you can look at the values in your table, or you can write the equation modeling each scenario and substitute in 6 for $x$ . For 6 bushels, Option A is better at $9.00.
To determine which scenario is better when she picks 12 bushels, you can extend the table or use the graph or equation. Option A is $18.00 for 12 bushels, and Option B is $5,314.41 for 12 bushels, so Option B is better.
To determine where Option B becomes the better option, you can extend the table, or you can graph the functions and find the $x$ -value where Option B surpasses Option A. You can also use problems 3 and 4 to help you. The number of bushels is more than 6 but less than 12. Option B is better if you are going to pick 7 or more bushels of apples.

Try it

Determining and Comparing Linear and Exponential Functions

Jayden has a dog-walking business. He has two plans. Plan 1 includes walking a dog once a day for a rate of $5 per day. Plan 2 also includes one walk a day but charges 1 cent for 1 day, 2 cents for 2 days, 4 cents for 3 days, and 8 cents for 4 days, and it continues to double for each additional day. Mrs. Maroney needs Jayden to walk her dog every day for two weeks. Which plan should she choose? Show the work to justify your answer.

Compare your answer:

Here is how to determine which of Jayden’s plans Mrs. Maroney should choose:

Plan 1 is a linear plan since it’s $5 per day. Plan 2 is doubling each day, so it is an exponential plan. Every other example has shown us that the exponential plan is going to be greater eventually. Since Mrs. Maroney needs Jayden for 2 weeks, or 14 days, we need to determine when Plan 2 becomes greater than Plan 1.

Plan 1 for 14 days would be:

$5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 5 \cdot 14 = $ 70$

Plan 2 for 14 days would be:

$.01 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = .01 (2)^{14} = $ 163.84$

Plan 1 would be better for Mrs. Maroney.