Lesson 5: Transformations of Exponential Functions

Property

For an exponential function $f(x)=a^x$, the graph can be translated:

1. Horizontal shift: The graph of $g(x) = a^{x-h}$ is the graph of $f(x)$ shifted $h$ units horizontally.
2. Vertical shift: The graph of $g(x) = a^x + k$ is the graph of $f(x)$ shifted $k$ units vertically. The horizontal asymptote also shifts to $y=k$.

Examples

• The graph of $g(x) = 2^{x-3}$ is the graph of $f(x) = 2^x$ shifted 3 units to the right. The point $(0, 1)$ on $f(x)$ moves to $(3, 1)$ on $g(x)$.
• The graph of $h(x) = 3^x + 5$ is the graph of $f(x) = 3^x$ shifted 5 units up. The horizontal asymptote moves from $y=0$ to $y=5$.
• To graph $k(x) = 4^{x+1} - 2$, take the graph of $f(x)=4^x$, shift it 1 unit to the left, and then 2 units down.

Explanation

Think of it as moving the entire picture of the graph. Adding or subtracting inside the exponent slides the graph left or right. Adding or subtracting outside the function moves it up or down, taking the horizontal asymptote with it.

Property

For horizontal translations $f(x) = a^{(x-h)}$, the direction of translation is opposite to the sign of $h$:
when $h > 0$, the graph shifts right by $h$ units;
when $h < 0$, the graph shifts left by $|h|$ units.

Examples

Property

When an exponential function $f(x) = a^x$ is vertically translated by $k$ units to form $g(x) = a^x + k$:
the range changes from $(0, \infty)$ to $(k, \infty)$ when $a > 1$;
the range changes from $(0, \infty)$ to $(k, \infty)$ when $0 < a < 1$.

Examples