If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Exponential graphs | Lesson

A guide to exponential graphs on the digital SAT

## What are exponential graphs?

In an exponential function, the output of the function is based on an expression in which the input is in the exponent. For example, $f\left(x\right)={2}^{x}+1$ is an exponential function, because $x$ is an exponent of the base $2$.
The graphs of exponential functions are nonlinear—because their slopes are always changing, they look like curves, not straight lines:
In this lesson, we'll learn to:
1. Graph exponential functions
2. Identify the features of exponential functions
You can learn anything. Let's do this!

## How do I graph exponential functions, and what are their features?

### Graphing exponential growth & decay

Graphing exponential growth & decaySee video transcript

### Using points to sketch an exponential graph

The best way to graph exponential functions is to find a few points on the graph and to sketch the graph based on these points.
To find a point on the graph, select an input value and calculate the output value. For example, for the function $f\left(x\right)={2}^{x}+1$, if we want to find the $y$-value when $x=1$, we can evaluate $f\left(1\right)$:
$\begin{array}{rl}f\left(1\right)& ={2}^{1}+1\\ \\ & =3\end{array}$
Since $f\left(1\right)=3$, the point $\left(1,3\right)$ is a point on the graph.
We need to use the points to help us identify three important features of the graph:
• What is the $y$-intercept?
• Is the slope of the graph positive or negative?
• What happens to the value of $y$ as the value of $x$ becomes very large?

#### The $y$‍ -intercept

Not only is the $y$-intercept the easiest feature to identify, it also helps you figure out the rest of the features.
To find the $y$-intercept, evaluate the function at $x=0$.
For example, the $y$-intercept of the graph of $f\left(x\right)={2}^{x}+1$ is:
$\begin{array}{rl}f\left(0\right)& ={2}^{0}+1\\ \\ & =2\end{array}$

#### The slope

An exponential function is either always increasing or always decreasing. If you have already evaluated $f\left(0\right)$, try evaluating $f\left(1\right)$.
• If $f\left(1\right)>f\left(0\right)$, then the slope of the graph is positive.
• If $f\left(1\right), then the slope of the graph is negative.
For $f\left(x\right)={2}^{x}+1$, since $f\left(0\right)=2$ and $f\left(1\right)=3$, we can conclude that the slope of the graph is positive because $3>2$.

#### The end behavior

End behavior is just another term for what happens to the value of $y$ as $x$ becomes very large in both the positive and negative directions. For the graph of an exponential function, the value of $y$ will always grow to positive or negative infinity on one end and approach, but not reach, a horizontal line on the other. The horizontal line that the graph approaches but never reaches is called the horizontal asymptote.
For $f\left(x\right)={2}^{x}+1$:
• As $x$ increases, $f\left(x\right)$ becomes very large. The value of $y$ on the right end of the graph approaches infinity.
• As $x$ decreases, $f\left(x\right)$ becomes closer and closer to $1$, but it's always slightly larger than $1$. The value of $y$ on the left end of the graph approaches, but never reaches, $1$.

#### Putting it all together

With the help of a few more points, $\left(-2,1.25\right)$, $\left(-1,1.5\right)$, and $\left(2,5\right)$, we can sketch the graph of $f\left(x\right)={2}^{x}+1$.
Note: if you're graphing by hand, it's more important to recognize that the value of $y$ will grow to positive infinity as $x$ increases than getting the graph exactly right! You can use the points you identified to establish a trend and sketch out the curve.
To graph an exponential function:
1. Evaluate the function at various values of $x$—start with $-1$, $0$, and $1$. Find additional points on the graph if necessary.
2. Use the points from Step 1 to sketch a curve, establishing the $y$-intercept and the direction of the slope.
3. Extend the curve on both ends. One end will approach a horizontal asymptote, and the other will approach positive or negative infinity along the $y$-axis.

Example: Graph $f\left(x\right)=12\cdot {\left(\frac{1}{2}\right)}^{x}$.

### Try it!

try: find points on an exponential graph
$f\left(x\right)={2}^{x}-1$
Identify points on the graph of the exponential function above and completing the table below.
$x$$f\left(x\right)$
$-3$$-0.875$
$-1$
$0$
$1$$1$
$2$

try: describe an exponential graph
$f\left(x\right)={2}^{x}-1$
Using the points from the previous question, complete the following statements about the graph of the exponential function above.
The $y$-intercept of the graph is located at
.
As $x$ increases, $y$
.
As the value of $x$ decreases, the value of $y$ approaches, but never reaches,
.

## How do I identify features of exponential graphs from exponential functions?

### Graphs of exponential growth

Graphs of exponential growthSee video transcript

### Identifying features of graphs from functions

#### The basic exponential function

The most basic exponential function has a base and an exponent:
$f\left(x\right)={b}^{x}$
Let's consider the case where $b$ is a positive real number:
• If $b>1$, then the slope of the graph is positive, and the graph shows exponential growth. As $x$ increases, the value of $y$ approaches infinity. As $x$ decreases, the value of $y$ approaches $0$.
• If $0, then the slope of the graph is negative, and the graph shows exponential decay. In this case, as $x$ increases, the value of $y$ approaches $0$. As $x$ decreases, the value of $y$ approaches infinity.
• For all values of $b$, the $y$-intercept is $1$.
The graphs of $f\left(x\right)={1.5}^{x}$ and $f\left(x\right)={\left(\frac{2}{3}\right)}^{x}$ are shown below.

#### How do we shift the horizontal asymptote?

The $y$-value of every exponential graph approaches positive or negative infinity on one end and a constant on the other. We can change the constant value $y$ approaches by introducing a constant term to the function:
• For $f\left(x\right)={b}^{x}$, the value of $y$ approaches infinity on one end and the constant $0$ on the other.
• For $f\left(x\right)={b}^{x}+d$, the value of $y$ approaches infinity on one end and $d$ on the other.
The graphs of $f\left(x\right)={1.5}^{x}$ and $f\left(x\right)={1.5}^{x}+2$ are shown below.
• For $f\left(x\right)={1.5}^{x}$, as $x$ decreases, the value of $y$ approaches $0$.
• For $f\left(x\right)={1.5}^{x}+2$, as $x$ decreases, the value of $y$ approaches $2$.

#### How do we shift the $y$‍ -intercept?

We can change the $y$-intercept of the graph either by introducing a constant term (as above) or introducing a coefficient for the exponential term:
• For $f\left(x\right)={b}^{x}+d$, the $y$-intercept is $1+d$.
• For $f\left(x\right)=a\cdot {b}^{x}$, the $y$-intercept is $a\cdot 1=a$. In this form, $a$ is also called the initial value.
• For $f\left(x\right)=a\cdot {b}^{x}+d$, the $y$-intercept is $a+d$.
The graphs of $f\left(x\right)={1.5}^{x}+2$, $f\left(x\right)=2\cdot {1.5}^{x}$, and $f\left(x\right)=2\cdot {1.5}^{x}+2$ are shown below.
• For $f\left(x\right)={1.5}^{x}+2$, the $y$-intercept is $1+2=3$.
• For $f\left(x\right)=2\cdot {1.5}^{x}$, the $y$-intercept is $2\cdot 1=2$.
• For $f\left(x\right)=2\cdot {1.5}^{x}+2$, the $y$-intercept is $2+2=4$.

### Try it!

TRY: identify the features of an exponential graph without finding points
$f\left(x\right)=64\left(0.25{\right)}^{x}$
Consider the exponential function $f$ above. The $y$-intercept of its graph, or the initial value of the function, is
.
Because the base of the exponent, $0.25$, is less than $1$, the slope of the graph is
. As the value of $x$ increases by $1$, the value of $y$
.

## You turn!

Practice: match an exponential function to its graph
Which of the following is the graph of the function $y=4\cdot {0.5}^{x}$ ?

Practice: transform an exponential function
The graph of function $f$ is shown in the $xy$-plane above. Which of the following is the graph of $f\left(x\right)+2$ ?

## Things to remember

For $f\left(x\right)={b}^{x}$, where $b$ is a positive real number:
• If $b>1$, then the slope of the graph is positive, and the graph shows exponential growth. As $x$ increases, the value of $y$ approaches infinity. As $x$ decreases, the value of $y$ approaches $0$.
• If $0, then the slope of the graph is negative, and the graph shows exponential decay. In this case, as $x$ increases, the value of $y$ approaches $0$. As $x$ decreases, the value of $y$ approaches infinity.
• For all values of $b$, the $y$-intercept is $1$.
To shift the horizontal asymptote:
• For $f\left(x\right)={b}^{x}$, the value of $y$ approaches infinity on one end and the constant $0$ on the other.
• For $f\left(x\right)={b}^{x}+d$, the value of $y$ approaches infinity on one end and $d$ on the other.
To shift the $y$-intercept:
• For $f\left(x\right)={b}^{x}+d$, the $y$-intercept is $1+d$.
• For $f\left(x\right)=a\cdot {b}^{x}$, the $y$-intercept is $a\cdot 1=a$. In this form, $a$ is also called the initial value.
• For $f\left(x\right)=a\cdot {b}^{x}+d$, the $y$-intercept is $a+d$.