Exponential Functions

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 aa and bb are real numbers, a0a\neq 0, b>0b>0, and b1b\ne1, written in the form:
f(x)=abxf(x)=ab^x
The base of an exponential function is the value of bb in the function rule. To graph an exponential function, start by evaluating the function for several values of xx, and use corresponding values of xx and f(x)f(x) to write ordered pairs. Then plot the points, and connect them with a smooth curve.

Exponential functions may involve the base ee. The number ee is a mathematical constant that is an irrational number with an approximate value of 2.71828. Values of the exponential function f(x)=exf(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 xx results in a change in the value of the function by a factor of the base, bb.

Evaluating Exponential Functions with Different Bases

Value Base 33 Base 12\frac{1}{2} Base ee
xx
f(x)=3xf(x)=3^x
f(x)=(12)xf(x)=\left(\frac{1}{2}\right)^x
f(x)=exf(x)=e^x
1-1
31=133^{-1}=\frac{1}{3}
(12)1=2\left(\frac{1}{2}\right)^{-1}=2
e10.36788e^{-1}\approx0.36788
00
30=13^0=1
(12)0=1\left(\frac{1}{2}\right)^{0}=1
e0=1e^0=1
1
31=33^1=3
(12)1=12\left(\frac{1}{2}\right)^1=\frac{1}{2}
e12.71828e^1\approx2.71828
22
32=93^2=9
(12)2=14\left(\frac{1}{2}\right)^2=\frac{1}{4}
e27.38906e^2\approx7.38906

As the value of xx increases by 1, the value of each exponential function changes by a factor of its base. For example, as each xx-value increases by 1, the value of f(x)=3xf(x)=3^x changes by a factor of 3. The value of each exponential function is 1 when x=0x=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=cy=c, where cc 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)=bxf(x)=b^x, where b>0b>0 and b1b\ne1. The domain is all real numbers, and the range is y>0y>0. This means that the graph never touches or goes below the xx-axis, where y=0y=0. Using interval notation, the domain is (,)(-\infty, \infty), and the range is (0,)(0, \infty). The horizontal asymptote is y=0y=0. Two points on the graph are (0,1)(0,1) and (1,b)(1,b).

An increasing function is a function in which the yy-values increase as the xx-values increase. A decreasing function is a function in which the yy-values decrease as the xx-values increase. For f(x)=bxf(x)=b^x, the function is increasing if b>1b>1 and decreasing if 0<b<10<b<1.
Determining whether the graph of an exponential function in the form f(x)=bxf(x)=b^x is increasing or decreasing depends on the relationship between the xx- and yy-values and the value of bb. For example, the graph of f(x)=exf(x)=e^x is increasing because the yy-values increase as the xx-values increase and e>1e>1. The graph of f(x)=(12)xf(x)=\left(\frac{1}{2}\right)^x is decreasing because the yy-values decrease as the xx-values increase and 0<12<10\lt \frac{1}{2}\lt 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)=bxf(x)=b^{x} can be translated vertically kk units and horizontally hh units.
Vertical Translations Horizontal Translations
For the graph of f(x)=bxf(x)=b^x and k>0k>0:
  • The graph of f(x)+k=bx+kf(x)+k=b^x+k is translated up by kk units.
  • The graph of f(x)k=bxkf(x)-k=b^x-k is translated down by kk units.
For the graph of f(x)=bxf(x)=b^x and h>0h>0:
  • The graph of f(xh)=bxhf(x-h)=b^{x-h} is translated right by hh units.
  • The graph of f(x+h)=bx+hf(x+h)=b^{x+h} is translated left by hh units.

An exponential function f(x)=bxf(x)=b^x can be stretched or compressed by a factor of aa. It can also be reflected across the xx-axis or yy-axis.
Stretches and Compressions Reflections
For the graph of f(x)=bxf(x)=b^x and a>0a>0:
  • The graph of af(x)=abxaf(x)=ab^x is a vertical stretch of the graph of f(x)f(x) by a factor of aa if a>1a>1.
  • The graph of af(x)=abxaf(x)=ab^x is a vertical compression of the graph of f(x)f(x) by a factor of aa if 0<a<10\lt a\lt 1.
For the graph of f(x)=bxf(x)=b^x:
  • The graph of f(x)=bx-f(x)=-b^x is a reflection of the graph of f(x)f(x) across the xx-axis.
  • The graph of f(x)=bxf(-x)=b^{-x} is a reflection of the graph of f(x)f(x) across the yy-axis.

Step-By-Step Example
Graphing a Translation of an Exponential Function
Graph the given exponential function:
f(x)=3x+32f(x)=3^{x+3}-2
Step 1

Identify the parent function. Sketch its graph.

The base of the function is 3:
f(x)=3x+32f(x)={\color{Red}{3}}^{x+3}-2
So, the parent function is:
f(x)=3xf(x)={\color{Red}{3}}^x
Step 2

Identify and perform any stretches, compressions, or reflections.

The given function can be rewritten as:
f(x)=13x+32f(x)=1\cdot 3^{x+3}-2
When the parent function is multiplied by a constant, aa, the graph is stretched if a>1a>1 or compressed if 0<a<10<a<1. If a<0a<0, the graph is reflected across the xx-axis.

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

Solution

Perform the translations. If a value hh is added to the xx-value in a function, the graph is translated hh units to the left. If a value kk is subtracted from the entire function, the graph is translated down by kk units.

  • Since h=3h=3 and it is added, translate the parent function 3 units to the left.
  • Since k=2k=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)=23x+1f(x)=-2\cdot3^x+1
Step 1

Identify the parent function. Sketch its graph.

The base of the given function is 3:
f(x)=23x+1f(x)=-2\cdot{\color{Red}{3}}^{x}+1
So, the parent function is:
f(x)=3xf(x)=3^x
Step 2
Identify and perform any vertical stretches or compressions.
f(x)=23x+1=123x+1\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, aa, the graph is stretched if a>1a>1 or compressed if 0<a<10<a<1. If a<0a<0, the graph is reflected across the xx-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 xx-axis. So, the graph from Step 2 is reflected across the xx-axis.
Solution

Perform the translations. If a value hh is added or subtracted from the xx-value in a function, the graph is translated hh units to the left or right. If a value kk is added to the entire function, the graph is translated up by kk units.

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