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What Are Exponential Functions?

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When something increases or decreases by the same percentage over a fixed period, you have an exponential (percentage) growth. The exponential growth may be negative, meaning the graph will decrease to the right rather than ascending upwards as it otherwise would.

Theory

Exponential Functions

An exponential function is expressed in this way:

f (x) = a \cdot b^{x}

Notice that the variable is now the exponent. Both $a$ and $b$ are numbers. We call $a$ the start value and $b$ the growth factor.

When the value of $a$ in the function is positive, the graph looks like one of the two graphs below.

Rule

The Graph of an Exponential Function

The graphs of an increasing and a decreasing exponential function

$a$ is the $y$ -value when $x = 0$ , $b$ is the growth factor,

$0 < b < 1$ blue graph, $b > 1$ red graph.

In general, $b > 1$ gives you a fixed percentage increase, $0 < b < 1$ gives you a fixed percentage reduction and $b = 1$ gives no change. The number $b$ acts as a growth factor. The value of $a$ affects the sign of the function values.

Example 1

Suppose you deposit $$ 5000$ into a savings account. You get $3 %$ interest per year on this deposit. How much will you have in your account after 7 years?

This is an example of exponential growth. First you need to find the growth factor associated with a $3$ % increase:

\begin{array}{l} Growth Factor & = 1 + \frac{3}{100} \\ = 1 + 0.03 \\ = 1.03 \end{array}

Growth Factor = 1 + \frac{3}{100} = 1 + 0.03 = 1.03 .

The initial value is

$ 5000

and the growth factor is

1.03

. Since the money is in the account for 7 years, you must therefore multiply the growth factor by itself seven times. In mathematics, this is written as a power with 7 as the exponent.

You must calculate the following terms to find the amount of money you have after seven years:

5000 \cdot 1.0 3^{7} = 6149.37 .

Thus, you have $$ 6149.37$ in your account after seven years.

Example 2

You have the function $f (x) = 3 \cdot 2^{x}$ . This function intersects the $y$ -axis at $y = 3$ and grows exponentially. This form of growth is very powerful. References to this type of graph are also often used in everyday speech, when someone says something is experiencing “exponential” growth.

Example of increasing exponential function

Example 3

You have the function $f (x) = 3 \cdot 0 . 5^{x}$ . This intersects the $y$ -axis at $y = 3$ and decreases exponentially. This form of reduction is very powerful. It’s like the downward version of the previous example.

Example of decreasing exponential function

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Exponential Functions with Euler's Number

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