Exponential Growth and Decay

Explore initial value and base, decay factor, percent change and context.

Lesson 1 of 53 parts Practice Flashcards

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Initial Value and Base

Objective: Explain initial value and base and use it to solve or interpret a problem.

In y=A·b^t, A is the output at t=0 and b is the per-period multiplier. A positive b>1 gives growth when A>0. A because b⁰=1.

Worked examples

Try itExponential Growth and Decay — practice 1.1: Start at 5 and multiply by 4 each period. Find the value after 5 periods.Show answer

Step 1: Use yₜ=5·4^t. Step 2: At t=5, compute 4^5=1024. Step 3: Multiply by 5 to get 5120.

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Initial Value and Base

Objective: Explain initial value and base and use it to solve or interpret a problem.

In y=A·b^t, A is the output at t=0 and b is the per-period multiplier. A positive b>1 gives growth when A>0. A because b⁰=1.

Worked examples

Try itExponential Growth and Decay — practice 1.1: Start at 5 and multiply by 4 each period. Find the value after 5 periods.Show answer

Step 1: Use yₜ=5·4^t. Step 2: At t=5, compute 4^5=1024. Step 3: Multiply by 5 to get 5120.

Decay Factor

Objective: Explain decay factor and use it to solve or interpret a problem.

A base between zero and one gives repeated fractional retention; a 20% decrease keeps 80% each period. Additive loss per step is not constant. No, it subtracts a share of the changing amount.

Worked examples

Try itExponential Growth and Decay — practice 2.1: Start at 1536 and halve for 8 periods. What remains?Show answer

Step 1: Use the factor (1/2)^8. Step 2: Divide 1536 by 2^8=256. Step 3: The remaining amount is 6.

Percent Change and Context

Objective: Explain percent change and context and use it to solve or interpret a problem.

A rate r written as a decimal gives growth multiplier 1+r or decay multiplier 1−r; use consistent time units and explain the range where the model is reasonable. Monthly and yearly rates are not interchangeable without conversion.

Worked examples

Try itExponential Growth and Decay — practice 3.1: A value of 70 grows by 10%. What is the new value?Show answer

Step 1: Convert 10% to multiplier 1.10. Step 2: Multiply 70 by 1.10. Step 3: The new value is 77.

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