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So then, what’s the difference between exponential growth and exponential decay?

So, remember when we were talking about the exponential growth functions

And how we derived it as something like this:

Well, the difference is b

The growth functions have b as a whole number

But the decay functions have b as a fraction

Makes sense though right?

If you take a fraction to a power, it’s going to keep giving you a smaller fraction

Well, that’s what we call decay

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SO, DOES THIS MEAN THAT NOTHING REALLY CHANGES EXCEPT B?

Yes

We still have the same equation:

Where h moves the graph left or right

k moves the graph up or down, and

a makes the graph stretch or compress

Nothing else changes…..

Except how the graph looks

So to start, just like before, let’s look at one of the easier examples:

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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:

X	Y

-2

-1

So, as we can see, as x gets bigger

F(x) gets smaller substantially

We can also see that:

Y-Intercept: (0, 1)

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NOW THAT WAS A LITTLE DIFFERENT

Think of it as the opposite of the growth function

When x gets bigger, y gets smaller.

Now, like I said before, nothing changes.

If we manipulate h, like so:

The graph moves to the right

If we manipulate k, like so:

The graph moves up

And finally,

If we manipulate a, like so:

The graph stretches vertically

Again, all of the normal rules apply

It’s just the function is getting smaller instead of bigger

So, now that we know this, let’s try an example:

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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:

X	Y

-2

-1

And from this graph we can see:

Y-Intercept: (0, 6)

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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:

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Example 1

Tony purchased a brand new truck for $40,000. The dealer told him that the truck will decrease by 9.5% each year. What will the truck be worth in 20 years?�

Alright, so as we discussed, exponential decay isn’t much different than exponential growth.

The same can be said for word problems.

Again, all we need to do is figure out:

What the initial cost is

What our ratio is

And the time we’re talking about.

So in this example, my initial cost is $40,000

So a is:

a = 40000

r = .095

= $5,432.90

And the trick to finding r is looking for the word “per”

Well, in this case, it’s 9.5%

And the time is 20 years.

So:

t = 20

Now remember, Tony is losing money in this deal.

So, we’re not going to add for exponential decay

We’re going to subtract.

So:

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So what is e?

e is what we call Euler’s number (pronounced Oy-ler’s)

Essentially it’s the number that you get if you run the function:

It’s what we call an irrational number, meaning it’s like pi

It continues on forever, and never repeats

And just like pi, we approximate e to the number: 2.718

The reason this is useful, is because it’s used in what we call compound growth.

Essentially, e is in a number of calculations, and helps us figure out how long until something happens

Or, how much will happen in a certain amount of time.

I know this doesn’t make sense right now, but we will be getting into it soon enough

For right now, just let it settle that we use Euler’s number to find compound interest

But for right now, let’s look at how we graph e.

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So, knowing what e is, when we graph the function:

We get something like this:

Now, the reason this is so important is because of how the slope of this curve works.

Although this is a calculus thing

When you look at the slope of the curve at each point

The slope is equal to e

Where as when you look at something like:

The slope of each point will be the line: y = 2x

Not a number like e.

Again, don’t freak out too much if you’re not understanding completely

This is just an introduction to e.

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SO, DOES THE GRAPH OF E FOLLOW THE SAME BASIC RULES?

Yes

We still have the same equation:

Where h moves the graph left or right

k moves the graph up or down, and

a makes the graph stretch or compress

Nothing else changes…..

The only thing that does is how we find our reference points

Our references points will be under the form:

These points are the only two points you need to graph the graph essentially.

Well that, and the asymptote

So again, nothing for the graph really changes

Except that b is now equal to the number e

So to start, just like before, let’s look at a change:

Reference points:

(0,1) -> (h, a + k)

(1, e) -> (h + 1, a*e + k

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LOOKING AT THE CHANGES OF THE GRAPH OF E

So, to start with, let’s estimate e to 2.718, to keep things simple.

Now, if we have our original graph of:

Then we can see that graph moves to the right

If we manipulate k, like so:

The graph moves up

And finally,

If we manipulate a, like so:

The graph stretches vertically

Again, all of the normal rules apply

It’s just the function is the number e.

So, now that we know this, let’s try an example:

But this time we subtract an h from it:

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EXAMPLE 1:

Given the function:

Graph the function, and then identify:

The reference points

The domain and range

Okay, to start, let’s graph the function:

X	Y

The point we can start with is the first reference point

So:

-0.8

And since we know the asymptote will always be:

y = k

Our asymptote is:

y = 3

Then our graph should look like: