 # Graphing Exponential Functions

### Evaluating Exponential Functions The rule for an exponential function includes a variable as an exponent.
An exponential function is a function, where $a$ and $b$ are real numbers, $a\neq 0$, $b>0$, and $b\ne1$, written in the form:
$f(x)=ab^x$
The base of an exponential function is the value of $b$ in the function rule. To graph an exponential function, start by evaluating the function for several values of $x$, and use corresponding values of $x$ and $f(x)$ to write ordered pairs. Then plot the points, and connect them with a smooth curve.

Exponential functions may involve the base $e$. The number $e$ is a mathematical constant that is an irrational number with an approximate value of 2.71828. Values of the exponential function $f(x)=e^x$ can be approximated by using technology such as a graphing calculator.

For an exponential function, an increase of 1 in the value of $x$ results in a change in the value of the function by a factor of the base, $b$.

### Evaluating Exponential Functions with Different Bases

Value Base $3$ Base $\frac{1}{2}$ Base $e$
$x$
$f(x)=3^x$
$f(x)=\left(\frac{1}{2}\right)^x$
$f(x)=e^x$
$-1$
$3^{-1}=\frac{1}{3}$
$\left(\frac{1}{2}\right)^{-1}=2$
$e^{-1}\approx0.36788$
$0$
$3^0=1$
$\left(\frac{1}{2}\right)^{0}=1$
$e^0=1$
1
$3^1=3$
$\left(\frac{1}{2}\right)^1=\frac{1}{2}$
$e^1\approx2.71828$
$2$
$3^2=9$
$\left(\frac{1}{2}\right)^2=\frac{1}{4}$
$e^2\approx7.38906$

As the value of $x$ increases by 1, the value of each exponential function changes by a factor of its base. For example, as each $x$-value increases by 1, the value of $f(x)=3^x$ changes by a factor of 3. The value of each exponential function is 1 when $x=0$.

### Properties of Exponential Functions and Their Graphs The base of an exponential parent function determines whether the function is increasing or decreasing.

An asymptote is a line that a graph approaches as one of the variables approaches infinity or negative infinity. An asymptote is often shown on a graph as a dotted line. The graph of an exponential function has a horizontal asymptote. The equation for a horizontal asymptote has the form $y=c$, where $c$ is a real number.

A parent function is a function of a certain type that has the simplest algebraic rule. The parent exponential function is $f(x)=b^x$, where $b>0$ and $b\ne1$. The domain is all real numbers, and the range is $y>0$. This means that the graph never touches or goes below the $x$-axis, where $y=0$. Using interval notation, the domain is $(-\infty, \infty)$, and the range is $(0, \infty)$. The horizontal asymptote is $y=0$. Two points on the graph are $(0,1)$ and $(1,b)$.

An increasing function is a function in which the $y$-values increase as the $x$-values increase. A decreasing function is a function in which the $y$-values decrease as the $x$-values increase. For $f(x)=b^x$, the function is increasing if $b>1$ and decreasing if $0. Determining whether the graph of an exponential function in the form f(x)=bxf(x)=b^xf(x)=bx is increasing or decreasing depends on the relationship between the xxx- and yyy-values and the value of bbb. For example, the graph of f(x)=exf(x)=e^xf(x)=ex is increasing because the yyy-values increase as the xxx-values increase and e>1e>1e>1. The graph of f(x)=(12)xf(x)=\left(\frac{1}{2}\right)^xf(x)=(21​)x is decreasing because the yyy-values decrease as the xxx-values increase and 0<12<10\lt \frac{1}{2}\lt 10<21​<1.

### Transformations of Exponential Functions Transformations can be used to graph exponential functions.
A transformation is a change of the graph of a function. Transformations of exponential functions are performed in the same way as transformations of other parent functions. An exponential parent function of the form $f(x)=b^{x}$ can be translated vertically $k$ units and horizontally $h$ units.
Vertical Translations Horizontal Translations
For the graph of $f(x)=b^x$ and $k>0$:
• The graph of $f(x)+k=b^x+k$ is translated up by $k$ units.
• The graph of $f(x)-k=b^x-k$ is translated down by $k$ units.
For the graph of $f(x)=b^x$ and $h>0$:
• The graph of $f(x-h)=b^{x-h}$ is translated right by $h$ units.
• The graph of $f(x+h)=b^{x+h}$ is translated left by $h$ units.

An exponential function $f(x)=b^x$ can be stretched or compressed by a factor of $a$. It can also be reflected across the $x$-axis or $y$-axis.
Stretches and Compressions Reflections
For the graph of $f(x)=b^x$ and $a>0$:
• The graph of $af(x)=ab^x$ is a vertical stretch of the graph of $f(x)$ by a factor of $a$ if $a>1$.
• The graph of $af(x)=ab^x$ is a vertical compression of the graph of $f(x)$ by a factor of $a$ if $0\lt a\lt 1$.
For the graph of $f(x)=b^x$:
• The graph of $-f(x)=-b^x$ is a reflection of the graph of $f(x)$ across the $x$-axis.
• The graph of $f(-x)=b^{-x}$ is a reflection of the graph of $f(x)$ across the $y$-axis.

Step-By-Step Example
Graphing a Translation of an Exponential Function
Graph the given exponential function:
$f(x)=3^{x+3}-2$
Step 1

Identify the parent function. Sketch its graph.

The base of the function is 3:
$f(x)={\color{Red}{3}}^{x+3}-2$
So, the parent function is:
$f(x)={\color{Red}{3}}^x$
Step 2

Identify and perform any stretches, compressions, or reflections.

The given function can be rewritten as:
$f(x)=1\cdot 3^{x+3}-2$
When the parent function is multiplied by a constant, $a$, the graph is stretched if $a>1$ or compressed if $0. If $a<0$, the graph is reflected across the $x$-axis.

In this case, the value of $a$ is 1, so the graph of the parent function is not vertically stretched or compressed, and is not reflected across the $x$-axis.

Solution

Perform the translations. If a value $h$ is added to the $x$-value in a function, the graph is translated $h$ units to the left. If a value $k$ is subtracted from the entire function, the graph is translated down by $k$ units.

• Since $h=3$ and it is added, translate the parent function 3 units to the left.
• Since $k=2$ and it is subtracted, translate the parent function 2 units down.
Step-By-Step Example
Graphing a Transformation of an Exponential Function
Graph the given function:
$f(x)=-2\cdot3^x+1$
Step 1

Identify the parent function. Sketch its graph.

The base of the given function is 3:
$f(x)=-2\cdot{\color{Red}{3}}^{x}+1$
So, the parent function is:
$f(x)=3^x$
Step 2
Identify and perform any vertical stretches or compressions.
\begin{aligned}f(x)&=-2\cdot 3^x+1\\&=-1\cdot2\cdot 3^x+1\end{aligned}
When the parent function is multiplied by a constant, $a$, the graph is stretched if $a>1$ or compressed if $0. If $a<0$, the graph is reflected across the $x$-axis. In this case, the function is multiplied by both -1 and 2. The result of multiplying the parent function by 2 is a vertical stretch by a factor of 2.
Step 3

Identify and perform any reflections.

The result of multiplying the parent function by -1 is a reflection across the $x$-axis. So, the graph from Step 2 is reflected across the $x$-axis.
Solution

Perform the translations. If a value $h$ is added or subtracted from the $x$-value in a function, the graph is translated $h$ units to the left or right. If a value $k$ is added to the entire function, the graph is translated up by $k$ units.

• Since $h=0$, the parent function is not translated to the left or right.
• Since $k=1$ and it is added, translate the graph 1 unit up.