Unit 6: Exponents and Exponential Functions

6.2: Exponential Functions

What is an exponential function?

Definition: Exponential Function

An exponential function has the variable in the exponent.

$f\left(x\right) = a \cdot b^x$, where $a$ is some constant and $b$ is positive (and not $1$).

Evaluating Exponential Functions

Example: Evaluating Exponential Functions

If $f\left(x\right)=2 \cdot 3^{x}$, what is $f\left(4\right)$?

$\begin{aligned}f\left(4\right) &=2 \cdot 3^{4} \\{}&=2 \cdot 81 \\{}&=162 \end{aligned}$

This sort of problem is most easily done using the calculator:
.

2*3^4

162

x

Example: Evaluating Exponential Functions

If $f\left(x\right)=2 \cdot \left(\frac{1}{3}\right)^{x}$, what is $f\left(4\right)$?

$\begin{aligned}f\left(4\right) &=2 \cdot \left(\frac{1}{3}\right)^{4} \\{}&=2 \cdot \frac{{1}^{4}}{{3}^{4}} \\{}&=2 \cdot \frac{1}{81} \\{}&=\frac{2}{81} \end{aligned}$

This sort of problem is best done using the calculator. Be sure to ask for the answer as a fraction, otherwise you'll get a decimal:
.

2*(1/3)^4

2/81

2*(1/3)^4

2/81

x

Exponential Growth

Definition: Exponential Growth

If $f$ is an exponential function $f\left(x\right) = a \cdot b^x$, and $a>0$, and $b>1$, the function will grow.

In this case, $a$ is the initial value and $b$ is the growth rate.

Example: Population Growth

Compound Interest

Compound interest is usually seen in the context of credit cards, where the amount owed goes up all the time. The formula here is for interest compounded annually:

$A\left(t\right) = P\left(1+i\right)^t$ where:

Example: Compound Interest

If we borrow $\$15000$ at an annual rate of $5.2\%$, how much do we owe after 6 years?

$\begin{aligned} A\left(t\right) &= P\left(1+i\right)^t \\ A\left(6\right) &= 15000\left(1+0.052\right)^6 \\ {} &= 20332.26203 \end{aligned}$

We owe $\$20332.26$. And probably have been sued by now.

Exponential Decay

Definition: Exponential Decay

If $f$ is an exponential function $f\left(x\right) = a \cdot b^x$, and $a>0$, and $0 \lt b \lt 1$, the function will shrink.

In this case, $a$ is the initial value and $b$ is the rate of decay.

Example: Population Decline