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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 quantity starts at 100 and grows by a factor of 3 each period. Write A(t).

  1. Worked Example: First identify exactly what the question is asking: A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 100.
  5. The multiplier is 3.
  6. Check the result by substituting or estimating: the response should match 100(3^t) and make sense in the original problem.

Answer: 100(3^t)

Practice problems

1. A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).

Show solution
  1. Warm-up: First identify exactly what the question is asking: A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 100.
  5. The multiplier is 3.
  6. Check the result by substituting or estimating: the response should match 100(3^t) and make sense in the original problem.

Answer: 100(3^t)

2. A quantity starts at 150 and grows by a factor of 4 each period. Write A(t).

Show solution
  1. Warm-up: First identify exactly what the question is asking: A quantity starts at 150 and grows by a factor of 4 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 150.
  5. The multiplier is 4.
  6. Check the result by substituting or estimating: the response should match 150(4^t) and make sense in the original problem.

Answer: 150(4^t)

3. A quantity starts at 200 and grows by a factor of 2 each period. Write A(t).

Show solution
  1. Core Practice: First identify exactly what the question is asking: A quantity starts at 200 and grows by a factor of 2 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 200.
  5. The multiplier is 2.
  6. Check the result by substituting or estimating: the response should match 200(2^t) and make sense in the original problem.

Answer: 200(2^t)

4. For A(t) = 250(2^t), find A(3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For A(t) = 250(2^t), find A(3).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute t = 3.
  4. 2^3 = 8.
  5. A(3) = 250 * 8 = 2000.
  6. Check the result by substituting or estimating: the response should match 2000 and make sense in the original problem.

Answer: 2000

5. For A(t) = 50(2^t), find A(3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For A(t) = 50(2^t), find A(3).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute t = 3.
  4. 2^3 = 8.
  5. A(3) = 50 * 8 = 400.
  6. Check the result by substituting or estimating: the response should match 400 and make sense in the original problem.

Answer: 400

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