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Exponential Functions

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

An exponential function has the form y = a * b^x (or f(x) = a * b^x), where a is the initial value (y at x = 0) and b is the constant multiplier per unit of x. If b > 1, the function grows; if 0 < b < 1, it decays.

What you'll learn

Why it matters: Compound interest, population growth, radioactive decay, and viral spread all model with exponential functions.

Worked example

Problem. If f(x) = 3 * 2^x, find f(0), f(1), and f(2).

  1. f(0) = 3 * 2^0 = 3 * 1 = 3.
  2. f(1) = 3 * 2^1 = 6.
  3. f(2) = 3 * 2^2 = 12.

Answer: 3, 6, 12

Practice problems

1. If f(x) = 5 * 3^x, find f(0).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 5 * 3^x, find f(0).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 3^0 = 1, so f(0) = 5 * 1 = 5.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. If f(x) = 5 * 3^x, find f(2).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 5 * 3^x, find f(2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 3^2 = 9, so f(2) = 5 * 9 = 45.
  4. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

3. If f(x) = 2 * 4^x, find f(1).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If f(x) = 2 * 4^x, find f(1).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. f(1) = 2 * 4 = 8.
  4. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

4. For y = 100 * (1.05)^x, enter the initial value.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For y = 100 * (1.05)^x, enter the initial value.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The initial value is the a in y = a * b^x.
  4. Here a = 100.
  5. Check the result by substituting or estimating: the response should match 100 and make sense in the original problem.

Answer: 100

5. For y = 100 * (1.05)^x, enter the growth factor.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For y = 100 * (1.05)^x, enter the growth factor.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. The growth factor is the b in y = a * b^x.
  4. Here b = 1.05.
  5. Check the result by substituting or estimating: the response should match 1.05 and make sense in the original problem.

Answer: 1.05

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