Exponential Growth Functions

Objective

• Review over geometric sequences
• Go over how to graph a geometric sequence
• Go over domain, range, y-intercept, and asymptote of the graph
• Talk about transforming exponential graphs
• Do some examples
• Homework
• Go over some work problems
• Translate the word problems
• Do some examples
• Homework Day 2

SO THEN, WHAT’S A GEOMETRIC SEQUENCE?

Well, a geometric sequence deals with exponential functions instead of linear ones.

Basically, where an arithmetic sequence deals with adding to the former (recursive)

The geometric sequence deals with multiplying to the former (still recursive)

So, to figure out what to do, we need to use the method of finite differences.

THE METHOD OF FINITE DIFFERENCES

To use the method of finite differences, we need to first start with a sequence first:

Next, we need to add “levels” to it to figure out what the power is:

The way we use the levels, is we figure out what we need to add to the sequence to get the next number.

For example:

+2

+6

+18

+54

Now, since we didn’t find a pattern, we go to the next level:

Looking at level 2, it looks like we’re seeing a pattern here.

It would seem each number is being multiplied by 3.

So, this would mean our function is:

Now let’s check to see if this is true:

So our method works!

However, there is another way to do this:

FINDING THE RATIO

Another way to find the function we are looking for is to take the ratio.

The issue with using this way, is that you need to know for a fact that you are dealing with a geometric sequence and not an arithmetic one.

So to use the ratio method, you take one of the sequences terms

And divide it by its former term.

Like so:

So, our explicit rule is:

Now that we know how to find the geometric functions

Let’s look at a few examples:

EXAMPLE 1:

Determine the explicit and recursive rules for this sequence.

+6

+42

+294

+2058

So, looking at this we can see this is going to be a geometric sequence.

But, to really find out what we are missing let’s make sure to look at the levels to determine what the function is.

So level 1:

Well, that didn’t help much, so let’s continue:

Well, it would seem we’re multiplying by 7 each time.

Although this one is harder to see.

Either way, we can see that our explicit rule will be:

And our recursive rule will be:

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)

Dissecting some word problems

So now that we know how to graph the function, and thereby find the domain, range, y-intercept, and asymptote

Now we need to move on to working with word problems

Why?

Because life is a giant word problem, and you need to be able to solve them in the real world

So, without further ado, let’s just try an example:

Example 1

Tony purchased a rare guitar in 2000 for $12,000. Experts estimate that its value will increase by 14% per year. How much will the guitar cost in 10 years?�

So, now we need to dissect this.

Tony bought a guitar for $12,000

This would mean our a is 12000

a = 12000

We know that every year, the value of the guitar gains 14%

Does this mean he is only getting 14% a year?

Or does this mean he is getting an extra 14% per year?

This would be extra, right?

He’s making money in this deal

So, as we can see, our r (ratio) is 14

And we’re adding 1 to the total

So something like:

r = .14

Cost = a(1 + r)

But, this is only for one year

The cost would increase every year

So, knowing that it increases 14% each year this means if the first year it was $1000

Year 2 would be 1140

Year 3 would be 1299.6

And so on

Which means this is definitely exponential

So, our equation looks like it would be:

= $44,486.66

And that’s how we solve these word problems

So, the formula for exponential growth is:

Where:

a = the beginning cost

r = percentage gained

t = time it continues

So now what we need is to figure out these three variables and we can solve any word problem

So, without further ado:

EXAMPLE 2:

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