4.1: Exponential Functions

We now look at an entirely different function: the exponential.

Your success in this chapter depends on having mastered the exponent properties in Section 1.2.

Don't forget to brush up on it if you forgot!

exponential function
The exponential function with base $a$ has the form \[f(x) = a^x\] where $a > 0$ and $a \neq 1$.

Suppose $f(x) = 3^x$. Evaluate $f(0), f(1), f(2)$ and $f\left(\frac{1}{2}\right)$.

Graphs of Exponential Functions

Exponentials grow very quickly, as $x$ increases. Here's an example.

Graph $f(x) = 2^x$ by hand.

In general, there are two possible shapes for $f(x) = a^x$:

The exponential function $f(x) = a^x$ has two possible graph shapes:

$a > 1$ $0 < a < 1$

In general:

Domain: $\mathbb{R}$
Codomain (range): $(0, \infty)$
Asymptotes: $y = 0$

Sketch the graph of $g(x) = 1 - 2^x$.

Compound Interest

Exponentials appear when calculating interest, or money earned when you lend/borrow an entity money.

For example, when depositing money into a bank account, you are lending the bank money, and the bank rewards you by giving you a percentage of the money you lent (the interest).

We start with the simple case, where the interest is only applied once.

simple interest
Interest applied only once against the principal.

principal
The money invested/borrowed initially.

Suppose you deposit \$1,000 into a Certificate of Deposit (CD). The interest is 12% per year. Calculate the amount after one year if you have simple interest.

Deriving Compound Interest

Instead of looking at one single year, we can look at multiple years.

Good reasons for doing this: in Year 2, we can earn interest on the interest earned in Year 1!

We know that in Year 1, the amount is $P(1 + i)$.

In Year 2, $P(1+i)$ is the new principal, which interest needs to be applied on. So the amount is \[\underbrace{P(1+i)}_{\text{new principal}} \cdot \underbrace{(1+i)}_{\text{applying interest}} = P(1+i)^2\]

In Year 3, $P(1+i)^2$ is the new principal, so the amount is \[\underbrace{P(1+i)^2}_{\text{new principal}} \cdot \underbrace{(1+i)}_{\text{applying interest}} = P(1+i)^3\]

In general, for Year $k$:

The amount $P(1+i)^k$ is an exponential function of $k$!

In our scheme, $i$ the interest rate per year.

In practice, we want to be able to apply interest for any time period (for example, each month). So we let \[i = \dfrac{r}{n} \qquad \qquad k = n\cdot t \] where

$r$ is the annual interest rate
$n$ is the # of times interest is compounded per year
$t$ is the # of years.

Substituting our new definition of $i$ and $k$ into $P(1+i)^k$ completes the derivation of compound interest:

compound interest
Compound interest is calculated by \[A(t) = P\left(1 + \dfrac{r}{n}\right)^{nt}\] where

$A(t) = $ amount after $t$ years
$P = $ principal
$r = $ interest rate per year
$n = $ number of times interest is compounded per year
$t = $ number of years.

\$1,000 is invested into a CD at a interest rate of 12% per year. Find the amount after three years if interest is compounded annually, quarterly, and daily.

In practice, you don't want all these numbers $n, r, t$ flying around when making a financial decision.

We use the Annual Percentage Yield (APY) instead: the simple interest rate in one year which accounts for compounding.

APY tells you the % gain you made in one year.

Find the APY for an investment that earns interest at a rate of 6% per year, compounded daily.

Notice that the APY is higher than the yearly interest rate $r$.

Compounding earns interest on your interest. APY adds up $r$ and interest on our interest into one number.

In our example the "interest on the interest" is APY $ - $ r $=$ 6.183% $- $ 6% $=$ 0.183%.

By compounding daily we earned an extra 0.183% more interest than if we only applied interest once per year!