Exponential Growth and Decay

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

Lesson 1 of 43 parts Practice Flashcards

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Initial Value and Base

Objective: Explain initial value and base, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$y=A b^t$

Worked examples

Try itStart at 5 and multiply by 3 each period. Find the value after 7 periods.Show answer

Step 1: Use yₜ=5·3^t. Step 2: At t=7, compute 3^7=2187. Step 3: Multiply by 5 to get 10935.

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Initial Value and Base

Objective: Explain initial value and base, apply the rule, and justify relevant restrictions.

Formulas

$y=A b^t$

Worked examples

Try itStart at 5 and multiply by 3 each period. Find the value after 7 periods.Show answer

Step 1: Use yₜ=5·3^t. Step 2: At t=7, compute 3^7=2187. Step 3: Multiply by 5 to get 10935.

Decay Factor

Objective: Explain decay factor, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$y=A(1−r)^t$

Worked examples

Try itStart at 640 and halve for 7 periods. What remains?Show answer

Step 1: Use the factor (1/2)^7. Step 2: Divide 640 by 2^7=128. Step 3: The remaining amount is 5.

Percent Change and Context

Objective: Explain percent change and context, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

Growth multiplier=1+r

Worked examples

Try itAn amount grows by 7% in one period. What is the multiplier?Show answer

Step 1: A growth multiplier equals 1 plus the rate as a decimal. Step 2: Convert 7% to 7/100. Step 3: The multiplier is 107/100.

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