Graph transformations of exponential and logarithmic functions, identify asymptotes, domain, range, and key points.
| x | y | Note |
|---|
| Parameter | Effect on graph | Affects |
|---|---|---|
| a > 1 | Vertical stretch by factor a | Range (exp), key points |
| 0 < a < 1 | Vertical compression | Range (exp), key points |
| a < 0 | Reflect over x-axis, then stretch by |a| | Range flips (exp) |
| b > 1 | Exponential growth / log increasing | Direction of curve |
| 0 < b < 1 | Exponential decay / log decreasing | Direction of curve |
| h | Shift right h units (left if h < 0) | Domain (log), asymptote |
| k | Shift up k units (down if k < 0) | Range (exp), asymptote |
y = bˣ and y = log_b(x) are inverses
Exponential and logarithmic functions are inverse functions of each other. Their graphs are reflections of each other across the line y = x.
This means every point (a, b) on y = bˣ corresponds to a point (b, a) on y = log_b(x).
Asymptotes: Both function families always have exactly one asymptote. Exponential functions have a horizontal asymptote; logarithmic functions have a vertical asymptote. These asymptotes become each other's reflection across y = x.
Domain & Range swap: Exponential: domain = all reals, range = (k, ∞). Logarithmic: domain = (h, ∞), range = all reals.
One-on-one Algebra 2 tutoring builds the visual intuition for how a, b, h, and k each shift, stretch, and flip a curve — we work through your actual homework so the patterns stick.