Hi,
I’m going to explain exponential growth and exponential decay. First of all, we
should start with exponential functions. Exponential functions are functions
that change at a constant rate. Exponential functions are usually in form 
.
.
Meanwhile,
A cannot be equal to zero and B has to be greater than 0. When A is greater
than 0 and B is greater than 1, the function has exponential growth. When A is
greater than 0 and B is less than 1, the function has exponential decay. In the
equation above, A represents the initial value, B represents the growth rate
and x represents the time.
Exponential
growth/decay has many real world applications and is therefore very important
in today’s society. It can help us determine population growth, how much money
we get from interest and it can help determine the half-life of things such as
bacteria. These all have important real world implications, which is why we
should delve further into the matter.
Lets
looks at some examples of exponential growth. Suppose Joe makes $100,000 a year
as an investment banker. Every year, Joe’s salary will increase by 12 percent
annually. How would you write this formula? And what will Joe’s salary be in 8
years?
Joe
has an initial salary of $100,000, therefore, A would be 100,000. Meanwhile,
Joe’s salary increases by 12 percent annually. So B would equal 1.12. In order
to find B, you have to add 1 to the
growth rate. So it would be 1+.12 and that would give us 1.12 for B. Since we
want to know what Joe’s salary in 8 years will be, X=8. Thus, the formula for
this equation would be Y= 100,000(1.12)^8 and his salary in 8 years would
be $247,596. We also know this is an
example of exponential growth because B>1.
The
following video is a cool illustration of exponential growth.
For
exponential growth/decay functions, I generally use two tricks. First, If I
know the growth rate is more than 1, the function is exponential growth. That
also helps you know how the graph of the function will look. The above graph (above link, I couldn't paste the graph) is
an example of exponential growth. If the growth factor is less than 1, I know
it will automatically know it will be exponential decay. The graph of an
exponential decay function will look very different from the graph of
exponential growth. The graph below (the link below, the graph wouldn't show on the blog) represents exponential growth. The initial
value of both graph are very different to help show the impact of the growth
factor. In the exponential growth graph, the initial value is relatively low
and over time the value goes up because there is growth. In contrast, the
exponential decay graph has a very high initial value but because there is a
decay rate the value decreases over time.
The
other trick has to deal with the growth factor. If I’m already given a percent,
It seems easier for me. If the percent is over 100 it would be similar to
getting a b value over 1 and I would know there is exponential growth.
Likewise, if the percent is 70 for example, I know there is a decay rate
because If I covert 70 percent to a decimal, I would get .7.
One
last example
After
Mary bought a car, she realized the value of her car would steadily depreciate.
She bought her car for $45,000 and her car will depreciate by 10 percent
annually. How much would her car be worth in three years?
In the given example, we are told that the car
will depreciate by 10 percent annually and we have an initial value of 45,000.
The tricky part of this equation is that they say it will depreciate by 10
percent annually. If we convert that to a decimal we get .1 the next step would
be to subtract 1 by .1 which will give us .9. This is because after one year,
the car will be worth 90% of what it was originally worth. Therefore the growth
factor in this equation would be .9 which means its exponential decay. Therefore, the
equation of this function would be f(x)= 45,000(.9)^3 and that would give Mary
a value of $32,805 after three years.
