Lesson 61: Exponential Functions and Graphs
An exponential function is a function in which the variable appears in the exponent: f(x) = a·b^x, where a ≠0 and b > 0, b ≠1. Exponential functions model growth and decay and have distinctive graph shapes.
Key Points About Exponential Graphs
- If b > 1 → Exponential growth, graph increases as x increases.
- If 0 < b < 1 → Exponential decay, graph decreases as x increases.
- The y-intercept is always at (0, a).
- The horizontal asymptote is y = 0 unless a transformation shifts it.
- Graphs never cross the x-axis; y is always positive for standard exponential functions.
Examples
Example 1: f(x) = 2^x → Growth
- y-intercept: (0,1)
- As x increases, y doubles each step
- Graph rises steeply to the right
Example 2: g(x) = (1/3)^x → Decay
- y-intercept: (0,1)
- As x increases, y decreases
- Graph approaches y = 0 as x → ∞
Example 3 (Transformation): h(x) = 3·2^x + 1
- Vertical stretch by 3, shifted up by 1
- Horizontal asymptote at y = 1
Practice Problems
- Identify if f(x) = 5^x represents growth or decay.
- Graph g(x) = 0.5^x, identify asymptote and y-intercept.
- Graph h(x) = 2·3^x − 2, label transformations.
- Find y-intercept of f(x) = 4·(1/2)^x + 3.
- Describe the long-term behavior of f(x) = 2·(3/4)^x.