Evaluating Exponential Functions
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 
$3^1=3$ 
$\left(\frac{1}{2}\right)^1=\frac{1}{2}$ 
$e^1\approx2.71828$ 
$2$ 
$3^2=9$ 
$\left(\frac{1}{2}\right)^2=\frac{1}{4}$ 
$e^2\approx7.38906$ 
Properties of Exponential Functions and Their Graphs
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 $0<b<1$.Transformations of Exponential Functions
Vertical Translations  Horizontal Translations 

For the graph of $f(x)=b^x$ and $k>0$:

For the graph of $f(x)=b^x$ and $h>0$:

Stretches and Compressions  Reflections 

For the graph of $f(x)=b^x$ and $a>0$:

For the graph of $f(x)=b^x$:

Identify the parent function. Sketch its graph.
The base of the function is 3:Identify and perform any stretches, compressions, or reflections.
The given function can be rewritten as:
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.
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.
Identify the parent function. Sketch its graph.
The base of the given function is 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.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.