Compound Interest

Objective

• Review over exponential growth function
• Review over how to graph exponential growth functions
• Go over compound interest
• Go over the different kinds of compound interest
• Do some examples
• Homework

So why are we reviewing over Geometric Sequences?

Because we’ve actually been going over exponential functions

But those exponential functions are the explicit functions of the geometric sequences we’ve been dealing with.

Regardless, we can use what we’ve learned to explore how exponential growth functions work.

Let’s start with looking at one of the easier functions that we’ve dealt with:

So, to start off understanding exponential growth, let’s look at a graph of one.

We’ll start with one of the easier ones we can work with:

-2

-1

0

1

1

2

2

4

3

8

So, as we can see, as x gets bigger

F(x) increases substantially

We can also see that:

Y-Intercept: (0, 1)

That’s great, but how does that help us?

It actually helps us quite a bit

See, now that we have a parent function to work with

We can graph things like:

And we can see how it changes!

-2

-1

0

1

2

1

3

2

So, as we can see from our previous graph

This graph moved two units to the right!

So, we found h, but if there a k?

Absolutely there is!

To find it, let’s try something like:

-2

-1

0

1

2

6

3

10

So, as we can see from our previous graph

This graph moved up two units!

NOW LET’S TALK ABOUT A

So, we have our h and now our k

But what about k?

Well, instead of going through the points (since I know you know how to do that)

I’m just going to show you the completed graphs so you can see the difference

So, to start, if our equation is:

Then if a is negative, we see a reflection about the x-axis

For example, looking at what we had before with:

Now let’s look at:

As you can see

When a became negative

The graph sloped down

Or another way to say that is

It reflected about the x-axis

SO WHAT HAPPENS IF A > 1?

So we know that happens when a is negative

But what about if it’s bigger than 1?

Well then, just like with any other graph, it stretches vertically

Again, let’s take our parent function:

And multiply it by an a that is bigger than 1

Let’s say, 4:

As you can see

When a is bigger than 1

The graph stretched upwards

SO WHAT HAPPENS IF 0 < A < 1?

So we know that happens when a is negative or if it’s bigger than 1

But what about if it’s less than 1?

Well then, just like with any other graph, it compresses vertically

Again, let’s take our parent function:

And multiply it by an a that is smaller than 1

Let’s say, 1/4:

As you can see

When a is less than 1

The graph is compressed

So now that we know how these graphs change

Let’s look at an example:

Example 1:

Graph the following function and identify the domain, range, y-intercept, and asymptotes:

So, first things first, let’s get a few points

Then we can figure the rest out.

So:

-2

-1

2

0

1

And from this graph we can see:

Y-Intercept: (0, -2)

So, why are we going over exponential growth?

Because we’re taking a quick turn, and talking about real life situations with compound interest.

So, compound interest, when done in your favor, gives exponential growth.

However, that’s only if you’re the one that’s getting the interest.

If you owe the interest, it’s a different story.

So, let’s get into it.

What we know already

So, believe it or not, we already know something about compact interest.

We know that to find it, we need the formula:

This is used when the interest of whatever it is, only happens once (like per year)

So, for example:

EXAMPLE 1:

John researches a pokémon card he found, and finds that it is currently worth $325.�However, it is supposed to increase in value 11% per year.

If he holds onto the card for 30 years, how much will the card be worth? �

Alright, so a few things

First, we need to know what the initial value is

Which in this case is $325

So:

a = $325

Next, we need to find our r.

However, a trick is to look for the word “per”

When you see “per”, the number before that is usually your r

So, in this case:

r = .11

And of course, our time is in years

So in this case:

t = 30

So now, we plug all of this into our equation, and we get:

= $7,439.99

SO ISN’T THIS THE SAME EXAMPLE AS WE HAD FOR EXPONENTIAL GROWTH?

It is!

But that’s because compound interest causes exponential growth.

Now, there are a few other times that compound interest needs another formula.

For example, what happens when the compound interest isn’t yearly?

What if it’s semiannually?

Or monthly? Or daily?

That’s where we start to use another type of formula.

OTHER THAN ANNUALLY COMPOUND INTEREST FORMULA

So, if you have compound interest that isn’t annually, then we need to come up with a different sort of formula that accounts for the difference in time.

Now, it’s still going to be the same basic idea

But with a little twist instead.

Remember, our annual formula is:

However, to figure out a different type of time, we actually need to divide our ratio by the time

And take the ratio plus 1 to the new time’s power

So, our new formula would look something like:

Where n is the new type of time.

This sounds way more complicated than it is, so to show this, let’s try an example:

EXAMPLE 1:

A person invests $1200 in an account that earns 2% interest compounded per quarter.�Find when the value of the investment reaches $1500 �

Alright, so a few things

First, we need to know what the initial value is

Which in this case is $1200

So:

a = $1200

Next, we need to find our r.

However, a trick is to look for the word “per”

When you see “per”, the number before that is usually your r

So, in this case:

r = .02

Now, the issue is that we don’t know what t is.

But we do know that n = 4 since the new time is quarterly:

n = 4

So now, we plug all of this into our equation, and we get:

Now, they want us to figure out when the value will reach $1500

So, what we do is graph our cost equation

And then graph y = 1500

Wherever they cross, that’s our answer.

(Use geogebra to do this!)

So:

Since our point is (11.2, 1500)

Then our answer is 11.2 years!

Interest Compounded Continuously

Now this is where it gets weird

If you have interest compounded continuously, then our formula is:

The way we get this is by taking our other equation:

And letting some arbitrary variable m = n/r

We do this because, if m = n/r, then 1/m = r/n

So:

And since:

m = n/r

r* * r

Then:

mr = n

And our equation becomes:

But since:

Substituting in e, we get:

Example 2:

A person invests $5000 in an account that earns 3.5% annual interest compounded�continuously. Find when the value of the investment reaches $12,000. �

Alright, so a few things

First, we need to know what the initial value is

Which in this case is $5000

So:

a = $5000

Next, we need to find our r.

In this case, we know it by its percentage

So:

r = .035

And we don’t know t, so let t = x

t = x

Now, they want us to figure out when the value will reach $12000

So, what we do is graph our cost equation

And then graph y = 12000

Wherever they cross, that’s our answer.

(Use geogebra to do this!)

So:

Since our point is (25, 12000)

Then our answer is 25 years!

Then: