Exponential Decay Functions
Objective
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:
X | Y |
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-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!
X | Y |
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-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:
X | Y |
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-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:
X | Y |
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-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
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
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:
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
4
-1
2
0
1
1
2
3
So, as we can see, as x gets bigger
F(x) gets smaller substantially
We can also see that:
Y-Intercept: (0, 1)
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:
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
4
0
1
And from this graph we can see:
Y-Intercept: (0, 6)
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 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:
Example 2
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, the initial cost was $400,000
So a is:
a = 400000
r = .1
= $718.80
And the trick to finding r is looking for the word “per” (or something similar)
Well, in this case, it’s 10%
And the time is 60 years.
So:
t = 60
Now remember, John is losing money in this deal.
So, we’re not going to add for exponential decay
We’re going to subtract.
So:
John bought his house, brand new, for $400,000. However, since he bought it, the neighborhood he lives in has declined rapidly, and every year his house loses 10% of its value. Considering John never wants to move, how much will his house be worth in 60 years when it’s left to his kids in his will?
Don’t worry, this doesn’t really happen this drastic in real life
Because,
Yikes.