Exponential Growth Worksheet

Compound growth is one of those math concepts that sounds abstract right up until it's attached to a dollar figure. Once students see that the same equation modeling bacteria growth or population change also models a retirement account, a car losing value, or inflation eating away at a paycheck, the formula stops being an exercise and starts being something they'll actually use.

This worksheet builds exponential growth and decay from the ground up, starting with the general form and the difference between a growth rate and a decay rate, then moving straight into financial applications: an investment compounding at a fixed annual return, a car depreciating year over year, and an equation students have to read backward to find the rate buried inside it. That last skill — starting from y = abx and pulling out what the numbers mean — is the same move students need when they encounter a compounding formula anywhere outside the classroom, whether that's a savings account disclosure or a loan term.

It's a natural fit for any unit connecting algebra to personal finance, since it asks students to move fluently between "here's a situation, write the equation" and "here's an equation, tell me what situation it describes." Both directions matter. Plenty of students can plug numbers into a formula they're handed but stall out when asked to build the formula themselves, or to explain what a given rate actually means in context.

From the Rapunzl curriculum: This worksheet is drawn from Module 20: Financial Exponents, the unit where students connect exponential functions to real growth and decay in money.

Calculating Exponential Change

Remember that the b value in the general form y = abx represents the growth or decay of the initial value. If the b value is 1, then no growth or decay happens. If you are told that the initial value changes by a percent, then we can represent the b value as growth or decay where r is the decimal form of the percent:

Growth is 1 + r Decay is 1 - r

1. Write an equation that represents the following situation: Starting value of 125, 6% growth per year, over 10 years.
2. Write an equation that represents the following situation: Starting value of 200, 2.5% decay per year, over 5 years.

Now use what you know about writing exponential functions with percent growth or decay to complete the following problems.

1. Farid makes a $12,000 investment that yields an average yearly return of 4%. If he makes no additional investments, how much will his investment be worth in 15 years?
2. You purchase a car for $36,000. You did some research and found that this particular model of car depreciates in value by around 15% per year. What is the value of your car after 6 years?
3. You make an investment where the balance over time can be modeled by the equation y = 32000(1.035)x, where x represents the number of years since the investment started and y represents your total balance after x years.
   • a. What is the starting balance of your investment?
   • b. What is the rate of growth of your investment?
4. y = 100(0.96)x is an equation that can be used to represent the purchasing power of $100 after x years of inflation. What is the rate of inflation used to make this calculation?

Teacher Notes

Problems 1 and 2 are the on-ramp: students build the equation from a plain-language description before they're asked to do anything harder with it. Watch for the most common slip here, which is forgetting to convert the percent to a decimal before adding or subtracting it from 1.

Problems 3 and 4 apply the same structure to a real financial decision — an investment return and a depreciating asset — so this is a good place to pause and ask students what they'd actually do with each number. An investment growing at 4% a year and a car losing 15% a year are the same math moving in opposite directions, and naming that out loud helps the growth-versus-decay distinction stick.

Problems 5 and 6 flip the task: instead of building an equation, students have to read one and extract the starting value and the rate. This is usually where the class splits into two groups — students who can identify the rate quickly and students who need to rewrite 1 + r or 1 - r explicitly before it clicks. Giving the second group permission to write that intermediate step out, rather than doing it mentally, tends to close the gap fast.

Want the rest of the Financial Exponents unit — the guided notes and the sequence this activity fits into? Rapunzl's teacher portal has it organized and ready to assign.

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