Lesson 4: Transformations of Exponential and Logarithmic Functions

Property

The graph of $f(x) = a^x + k$ shifts the graph of $f(x) = a^x$ vertically $k$ units.
The graph of $f(x) = \log_a(x) + k$ shifts the graph of $f(x) = \log_a(x)$ vertically $k$ units.

• If $k > 0$, shift the graph vertically up $k$ units.
• If $k < 0$, shift the graph vertically down $|k|$ units.

Examples

Property

For exponential and logarithmic functions, horizontal translations follow these patterns:

• The graph of $f(x) = a^{(x-h)}$ shifts the graph of $f(x) = a^x$ horizontally $h$ units.
• The graph of $f(x) = \log_a(x-h)$ shifts the graph of $f(x) = \log_a(x)$ horizontally $h$ units.
• If $h > 0$, shift right $h$ units. If $h < 0$, shift left $|h|$ units.

Examples

Property

The coefficient $a$ in exponential and logarithmic functions affects vertical stretching and compression:

• For $f(x) = a \cdot b^x$ or $f(x) = a \cdot \log_b(x)$, the coefficient $a$ multiplies all y-values
• If $0 < |a| < 1$, the graph is compressed vertically (closer to x-axis)
• If $|a| > 1$, the graph is stretched vertically (farther from x-axis)
• If $a < 0$, the graph is also reflected across the x-axis

Examples

Property

For a function $f(x)$, the transformation $g(x) = f(ax)$ creates a horizontal stretch or shrink by a factor of $\frac{1}{a}$: