Lesson 47: Graphing Exponential Functions

New Concept

A function of the form $y = ab^x$ is an exponential function if $x$ is a real number, $a \neq 0$, $b > 0$, and $b \neq 1$.

What’s next

Next, you’ll explore the parent function $y=b^x$ by creating tables and graphing curves to see how these powerful functions behave.

A function of the form $y = ab^x$ is an exponential function if $x$ is a real number, $a \neq 0$, $b > 0$, and $b \neq 1$. The parent function of all exponential functions with base $b$ is $y = b^x$.

The function $y = 5^x$ shows exponential growth because its base $b=5$ is greater than 1.
The function $y = (\frac{1}{3})^x$ shows exponential decay because its base $b=\frac{1}{3}$ is between 0 and 1.
The expression $y = (-4)^x$ is not an exponential function because its base is negative.

Think of 'b' as a growth multiplier. The variable 'x' in the exponent is the secret sauce that makes the function value skyrocket or shrink incredibly fast. This is why we call it exponential growth! The base 'b' must be a positive number; otherwise, you'd get a wacky, disconnected graph that isn't a true function.

An asymptote is a line that a graph approaches as the value of a variable becomes extremely large or small. The line $y = k$ is a horizontal asymptote of a graph if $y$ approaches $k$ as $x$ increases or decreases without bound.

For the graph of $y = 4^x$, the line $y = 0$ is a horizontal asymptote as $x$ approaches negative infinity.
In the function $y = 4^x - 5$, the entire graph is shifted down 5 units, so the horizontal asymptote becomes the line $y = -5$.
For $y = (\frac{1}{4})^x$, the asymptote is also $y=0$, but the graph approaches it as $x$ gets large and positive.

Imagine an invisible electric fence your graph gets incredibly close to but can never, ever touch. For a basic exponential function like $y=b^x$, this invisible barrier is the x-axis ($y=0$). The graph just flattens out and runs alongside it forever as it goes to infinity in one direction, creating a clear boundary.

Given the parent function $y=b^x$, a transformed function $y = ab^{x-h} + k$ is stretched or compressed by 'a', shifted horizontally by 'h', and shifted vertically by 'k'.

The graph of $y = \frac{1}{5} \cdot 2^x$ is a vertical compression of $y = 2^x$ by a factor of one-fifth.
The graph of $y = -(1.5)^x$ is a reflection of $y = (1.5)^x$ across the x-axis.
The graph of $y = 2^{x-3} + 4$ is the graph of $y=2^x$ shifted 3 units right and 4 units up.

Think of transforming exponential functions like giving the parent graph, $y=b^x$, a complete makeover. The 'a' value acts like a funhouse mirror, stretching or squishing it. The 'h' value makes it do a side-shuffle left or right, and 'k' makes it hop up or down. These numbers let you move and reshape the basic curve.