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. In Exponential Functions and Sequences, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.
What you'll learn
- Recognize the form y = a * b^x for an exponential function
- Identify a as the initial value and b as the growth (or decay) factor
- Evaluate exponential functions at integer inputs
Worked example
Problem. If f(x) = 3 * 2^x, find f(0), f(1), and f(2).
- f(0) = 3 * 2^0 = 3 * 1 = 3.
- f(1) = 3 * 2^1 = 6.
- f(2) = 3 * 2^2 = 12.
- Connect the result back to Exponential Functions so the method and meaning are both clear.
Answer: 3, 6, 12
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. If f(x) = 5 * 3^x, find f(0).
Show solution
- 3^0 = 1, so f(0) = 5 * 1 = 5.
Answer: 5
2. If f(x) = 5 * 3^x, find f(2).
Show solution
- 3^2 = 9, so f(2) = 5 * 9 = 45.
Answer: 45
3. If f(x) = 2 * 4^x, find f(1).
Show solution
- f(1) = 2 * 4 = 8.
Answer: 8
4. For y = 100 * (1.05)^x, enter the initial value.
Show solution
- The initial value is the a in y = a * b^x.
- Here a = 100.
Answer: 100
5. For y = 100 * (1.05)^x, enter the growth factor.
Show solution
- The growth factor is the b in y = a * b^x.
- Here b = 1.05.
Answer: 1.05
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