Introduction

• In this lesson you will learn how to graph and evaluate exponential functions.
• Exponential functions are functions written in the form , where a is the base and is positive and , and x is a real number.

Mathematics involves counting and measuring. Some things can be counted by multiplying continually. For example, bacteria reproduce by splitting, doubling the number of bacterial cells. If there are 7 cells and a doctor is examining the bacteria, after a certain time, there will be 14 cells….. then 28…… then 56, etc. A function which models this is where 7 is the original amount of bacteria and t is the time it takes for the bacteria to reproduce (double in quantity). If t = 0 we have the original amount of bacteria: = = 7. If reproduction has happened 4 times we have . We examine formulas which involve x as an exponent and calculations related to those formulas.

Graphs of Exponential Functions

Example 1: Consider the following example:

Let’s graph the functions using a table of values. Remember that constructing a table of values means that numbers are substituted for x in the equation to find values for f(x). Remember also that f(x) is another name for y. Substituting –2, -1, 0, 1, and 2 for x in the given exponential equation, we find the following values for f(x):

x	f(x)
-2
-1
0	1
1	3
2	9

Plotting these points on an axes provides the following graph of the exponential function:

Notice that the graph from Example 1 and this graph increase to the right of the y-axis and decrease to the left. Both graphs, pass through the point (0, 1). The reason for that, remember, is that any non-zero real number raised to the 0 power has a value of 1. If both graphs are drawn on the same axes, it can be seen that although the two graphs are very similar, they are not exactly the same.

The graph of is the one whose distance becomes further away from the y-axis as it increases. That is because it increases at a slower rate than the other. The larger the real number used as the base, the faster its graph, values of , will increase.

Notice that the graph in Example 1 approaches the x-axis as the x values decrease (the graph going to the left). The x-axis, equation y = 0 is an asymptote. This is because is always positive. There is no value of x which can make negative or zero. As the values of x decrease, so do the values of , but there is a limit which the graph will never reach, the x-axis.

Example 2: Consider the following example:

Using the same values for x, let’s graph this exponential function whose exponent is negative. Coordinates of several points are shown in the table below.

x	f(x)
-2	9
-1	3
0	1
1
2

The graph of this function is decreasing since it begins higher on the left side of the y-axis and continuously decreases in value as it moves toward and crosses over the y-axis. The graphs of functions in the form will be decreasing.

Summary Questions:

Complete these three final questions to help you summarize your understanding of the important facts presented in this lesson concerning properties of exponential function graphs.