Section 6.2: Graphing Exponentials
This section explains how to graph exponential functions, identifying asymptotes, intercepts, and growth or decay behavior.
Example 1: Graphing Growth
Graph \( f(x) = 2^x \). Identify the y-intercept and horizontal asymptote.
Step 1: Y-intercept at x = 0 → \( f(0) = 1 \)
Step 2: Horizontal asymptote: y = 0
Step 3: Plot a few points: x = 1 → 2, x = 2 → 4, x = -1 → 1/2
Example 2: Graphing Decay
Graph \( g(x) = (1/3)^x \). Identify the y-intercept and asymptote.
Step 1: Y-intercept at x = 0 → \( g(0) = 1 \)
Step 2: Horizontal asymptote: y = 0
Step 3: Plot points: x = 1 → 1/3, x = 2 → 1/9, x = -1 → 3
Practice Problems
- Graph \( f(x) = 3^x \). Identify intercept and asymptote.
- Graph \( h(x) = (1/2)^x \) and describe its behavior as x → ∞ and x → -∞.
- Graph \( f(x) = 5 \cdot 2^x \). Find the y-intercept and horizontal asymptote.
- Graph \( g(x) = 2^{-x} \). Identify growth or decay and asymptote.
- Graph \( f(x) = 3^{x-2} \). Find the y-intercept and horizontal asymptote.