Exponential Functions Worksheet

Exponential functions show up on every state math test, and most worksheets teach them the same way: give students a formula, then hand them a table of x-values to plug in. Students learn to execute the formula without ever having to build one. This worksheet flips that order. Instead of applying a given equation, students have to write the equation themselves, starting from a plain-language description of a real situation.

That shift matters because writing an exponential function requires students to identify two things a table never asks for: the starting value and the growth or decay factor. The worksheet opens with three warm-up problems built around doubling — start with 5 and double it three times, then nine times, then one hundred times — which forces students to notice that repeated doubling is multiplication by 2 raised to a power, not addition. From there, students move to interpreting two fully-formed exponential functions, pulling out the starting value, the growth or decay factor, and whether the function is growing or shrinking, before evaluating it at a given input.

The second half is where the worksheet earns its place in a finance-adjacent unit. One problem asks students to model Pinocchio's nose doubling with every lie, which is a low-stakes way to practice writing an exponential equation from a word problem before the numbers get more serious. The next problem raises the stakes for real: a $10,000 investment that has historically doubled every seven years. Students write the equation and then project its value fifteen years out, which is compound growth in miniature, without ever using the word "compound interest."

This works well as a short, self-contained activity — 20 to 25 minutes is enough for most classes to get through all seven problems, with time left over to discuss why the investment problem and the Pinocchio problem use the exact same mathematical structure even though one is silly and one is real money.

This activity is from Module 20 of the Rapunzl curriculum, Financial Exponents.

Writing Exponential Functions

Through this activity, students will learn to write exponential functions by examining real-world examples of exponential growth and decay. By the end of this exercise, students will be able to create exponential functions that accurately represent different scenarios.

1. If you were asked to start with the number 5 and double it three times, how would you write this problem using math notation?
2. If you were asked to start with the number 5 and double it nine times, how would you write this problem using math notation?
3. If you were asked to start with the number 5 and double it one hundred times, how would you write this problem using math notation?
4. Given the exponential function y = 3(5)x
   • a. State the starting value:
   • b. State the growth/decay factor:
   • c. Is the function growing or decaying?
   • d. What is the result if x = 3?
5. Given the exponential function y = 32(0.25)x
   • a. State the starting value:
   • b. State the growth/decay factor:
   • c. Is the function growing or decaying?
   • d. What is the result if x = 5?
6. Everytime Pinocchio lies, his nose doubles in size. His nose is 1.5 inches long before he has told any lies.
   • a. Write an equation that represents this situation where x is the number of lies and y is the size of Pinochhio’s nose after x lies.
   • b. Use your equation to calculate how long Pinocchio’s nose will be after 6 lies.
7. You make a $10,000 investment that has historically doubled every 7 years.
   • a. Write an equation that represents this situation where x is the number of years and y is the investment value after x years.
   • b. Use your equation to calculate the value of your investments after 15 years.

Teacher Notes

Problems 1 through 3 are the ones to watch closely. Students who get the doubling pattern right for three doublings often revert to repeated addition once the number of doublings jumps to nine or one hundred, because the answer stops being something they can count out on their fingers. That's the exact moment they need to see 2 raised to a power instead of 2 added repeatedly, so don't let those two warm-up problems feel like throwaways.

Problems 4 and 5 are best done side by side on the board, since one is growth and the other is decay. Students frequently mislabel a decay function as growth when the base is written as a decimal like 0.25 instead of as a percentage decrease, so ask them to explain in words what 0.25 means before they commit to an answer.

The Pinocchio and investment problems are meant to be discussed together. Once students see that a joke about a lying puppet and a real $10,000 investment use the identical equation structure, the abstraction of "exponential function" stops being abstract. Ask students directly what's different between the two situations and what's the same — the answer is nothing about the math changes, only the story around it.

The answer key for this worksheet, along with the rest of the activities in this unit, is in the Rapunzl teacher portal.

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