Exponential Functions and Their Graphs

The general form: f(x) = a cdot b^x + k where: - a: vertical stretch (initial multiplier) - b: base (growth/decay factor) - b > 1: exponential growth (graph rises steeply) - 0 < b < 1: exponential decay (graph falls toward zero) - k: horizontal asymptote (y = k) The y-intercept is f(0) = a cdot b^0 + k = a + k.

| Feature | Growth (b > 1) | Decay (0 < b < 1) | |---|---|---| | Direction | Increasing | Decreasing | | Shape | Rises steeply to the right | Falls toward asymptote | | Asymptote | y = k (approached on the left) | y = k (approached on the right) | | Y-intercept | (0, a + k) | (0, a + k) | | Domain | All real numbers | All real numbers | | Range | y > k (if a > 0) | y > k (if a > 0) | Critical di…

Example: Describe the graph of f(x) = 3 cdot 2^x - 1.