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Exponential Growth

A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.

Exponential growth uses a repeated multiplier greater than 1. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Exponential and logarithmic models describe growth, decay, sound, pH, finance, and scientific scales.

Worked example

Problem. A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).

  1. Worked Example: First identify exactly what the question is asking: A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Use A(x) = (starting value)(multiplier^x).
  4. The starting value is 30 and the multiplier is 3.
  5. So A(x) = 30(3^x).
  6. Check the result by substituting or estimating: the response should match 30(3^x) and make sense in the original problem.

Answer: 30(3^x)

Practice problems

1. A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).

Show solution
  1. Warm-up: First identify exactly what the question is asking: A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Use A(x) = (starting value)(multiplier^x).
  4. The starting value is 30 and the multiplier is 3.
  5. So A(x) = 30(3^x).
  6. Check the result by substituting or estimating: the response should match 30(3^x) and make sense in the original problem.

Answer: 30(3^x)

2. For A(x) = 40(2^x), find A(3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: For A(x) = 40(2^x), find A(3).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute x = 3.
  4. 2^3 = 8.
  5. A(3) = 40 × 8 = 320.
  6. Check the result by substituting or estimating: the response should match 320 and make sense in the original problem.

Answer: 320

3. A quantity increases by 200% each period. What multiplier is used?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A quantity increases by 200% each period. What multiplier is used?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. A percent increase adds onto the original 100%.
  4. 100% + 200% = 300% = 3 as a decimal.
  5. So the repeated multiplier is 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. For A(x) = 60(2^x), find the initial value A(0).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For A(x) = 60(2^x), find the initial value A(0).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute x = 0.
  4. 2^0 = 1, so A(0) = 60 × 1.
  5. The initial value is 60.
  6. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

5. Does A(x) = 20(3^x) model growth or decay?

Choices: growth · decay · neither — it is constant · decay then growth

Show solution
  1. Compare the base 3 with 1.
  2. Because 3 > 1, each period multiplies by more than 1.
  3. A base greater than 1 means exponential growth.

Answer: growth

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