Lesson 1: Key Features of Exponential Functions

Property

For $f(x) = ab^x$, with $a > 0$:

1. If $b > 1$, the function is increasing.
2. If $0 < b < 1$, the function is decreasing.
3. The $y$-intercept is $(0, a)$. There is no $x$-intercept.
4. The $x$-axis is a horizontal asymptote for the graph.

Examples

Property

For an exponential function $f(x) = b^x$:

• The graph always passes through the point $(0, 1)$, which is the y-intercept.
• The x-axis ($y=0$) is a horizontal asymptote, meaning the graph gets infinitely close but never touches it.
• If the base $b > 1$, the function is always increasing (representing exponential growth).
• If $0 < b < 1$, the function is always decreasing (representing exponential decay).

Examples

• The graph of $f(x) = 3^x$ is an increasing function. It passes through $(0, 1)$ and rises sharply to the right as $x$ increases.

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.