Activity
Consider the functions and .
1. Complete the table of values for the functions and .
1 | ||
10 | ||
50 | ||
100 | ||
500 |
Compare your answer:
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:
is growing faster for the values of in the table, but as increases, starts to grow faster. So as increases, may eventually grow faster than .
3. Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing by 100 a few times and record the function values in the table.
Compare your answer:
When is 600, and , so takes a larger value. When is 700, and , so still takes a larger value. But when is 800, and , so takes a larger value.
Are you ready for more?
Extending Your Thinking
Consider the functions and . While it is true that , for example, it is hard to check this using mental math. Find a value of for which properties of exponents allow you to conclude that without a calculator.
Compare your answer:
Your answer may vary, but here are some samples: One example is . We have , whereas , which is certainly smaller than .
Self Check
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 represents the bushels of apples picked and represents the total amount she earns based on 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 Meaning of Parameters and is the slope of the line or the constant rate of change; is the -intercept or the value at . is the -intercept or the value when ; is the base or the constant quotient of change. Example Rule for Finding from Starting at , to find , add 1.5 to . Starting at , to find , multiply by 3. 0 0 1 1.50 2 3.00 3 4.50 Linear Model Table0 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 . 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 -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
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:
Plan 2 for 14 days would be:
Plan 1 would be better for Mrs. Maroney.