Exponential Functions

Exponential functions, while similar to functions involving exponents, are different because
the variable is now the power rather than the base.

Before, we dealt with functions of the form

where the variable x was the base and the number was the power. If you notice, this
function is in the form of a quadratic. With exponential functions, they will be
similar to the form

where the number is the base and the variable is the exponent. Exponential function
will always have a positive number other than one as its base.

The definition of an exponential function is of the form

Now, how do the graphs of quadratics and exponentials differ? To graph an exponential
function, we just plug in values of x and graph as usual, but we need to remember
that if we plug in negative values for x, we need to put the quantity on the other
side of the fraction line.

Let’s graph the functions f(x) = x2 and g(x) = 2x.

Notice that to the left of the y axis, the graph approaches 0 but never touches
0. It may look like it, but these y values are so small that they are almost indistinguishable
from the x axis. To the right of the x axis it shoots up to infinity. If you have
ever heard of the term “exponential growth,” this is where it comes from. If you
ever hear about something doubling or tripling over a set increment, it is considered
exponential growth. Exponential functions tend to get very big very quickly, and
though they start out smaller than polynomial functions, they will always eventually
become bigger. Notice that the two functions meet at x = 2 and x = 4,
and then the exponential function becomes bigger than the quadratic. This is because
at x = 2, both functions are 22, and at x = 4, the functions are
also equal (42 = 24).

Exponential Growth and Decay

We have seen that exponential growth has the trend of starting out small and getting
bigger and bigger. Exponential growth and decay are common in nature, such as the
growth at the number of microorganisms in a culture or decay of sound vibrations.

Growth functions will have a positive integer raised to a positive power or a fraction
less than one raised to a negative integers. The following graphs will look the
same.

This is because when the fraction is raised to a negative power, the denominator
becomes numerator and the exponent becomes positive, so it is the same as exponential
growth!

Most exponential functions will look similar, except when we have exponential decay.
Decay functions will either be a positive fraction less than 1 raised to a positive
power or a positive integer raised to a negative exponent.

Let’s look at both the growth and decay graphs

There are two important things to notice. The decay graph is going in the opposite
direction of the growth graph. Also, no matter what exponential function, the value
of the function when x is 0 will always be 1. This is because any value raised to
0 is always 1.

Graph the following exponential function

With this function, we have a fraction less than one as the base. This must mean
it is exponential decay. We also have to operators – we are multiplying by 4 and
adding 3. Be careful with order of operations, because we need to deal with the
exponent first and then the operators.

We can see that the graph is indeed an exponential decay, and that it approaches
y = 3 but never touches it.

Solving for x

We should see that each exponential function has a horizontal asymptote where any
y value will never cross. This can be illustrated when we solve for x. Given the
equation

As we have seen in the exponents section in Algebra, we could see that when we set y equal to 2,
the exponents will be equal, and therefore x will be 1.

We can do this substitution for multiple y values

There is an easier way to solve for x by isolating it in terms of y. The only problem
is how. When we have addition, we subtract, and when we have multiplication, we
divide – but what do we do when we have an exponent? Well, we could raise it to
the reciprocal

This does not help us since we want to isolate x. We have learned that taking the
log is
an easy way to isolate an exponent. Let’s try it.

Here, we can plug in any y value and obtain our x value. We must be careful, because
we cannot take the log of any value less than or equal to 0. Let’s try a harder
example

We would go about this as we would we any other equation, treating the term with
the exponent as a variable until we have to deal with it.

This is a bit of a mess, but it does the trick! We have successfully isolated x
and can find any coordinate of the equation.

In general

Compound Interest

In finance, exponential functions are prevelent in dealing with calculating interest.
The compound interest formula is a very important exponential equation.

Compound Interest Formula

Where A is the ending amount, P is the beginning value, or principle
value, r is the interest rate (usually a fraction), n is the number
of compoundings a year, and t is the total number of years. We will see that
this formula simplifies to the exponential functions we are accustomed to.

Regarding n, if interest is compounded once a year, it would be considered
annually and n would be 1. If twice a year, it would be considered semi-annually
and n would be 2 (similarly, quarterly would be 4, monthly would be 12, and so on).
Since the interest rate is expressed in years, the time must be expressed in years
as well.

Suppose the interest rate is 4% compounded monthly, and let the initial investment
amount be $800. What is the ending amount after 10 years?

This is the form of an exponential function with base 1.08.

Suppose you want to know how many years until you have 900 dollars, how many years
will it take?

It would take about 3 years. By varying the frequency in which the interest is compounded or the rate, the interest
can be changed dramatically. Though this formula is important for managing money
and calculating interest given a bank’s interest rates and how many times it is
compounded yearly, what if we compounded it continuously? In other words, what if
we took the time t to infinity?

The Natural Exponential Function f(x) = ex

The value e is a mathematical constant that was discovered from the compound
interest problem. We discussed compounding interest at different increments per
year, but what if we keep going?

as we compound in smaller increments, our output yields the value of e.

Similar to pi, the value of e is irrational. Approximated to two decimal places,
it is equal to 2.72. The function f(x) = ex is a unique
exponential function because the y value is always equal to the rate of change of
the function at that point. No other function has this trait. This is studied further
in calculus when we study
rates of change.

At y = 7.39, the slope is also 7.39

We saw that if we compounded our interest to an infinite amount of increments, we
get the value of e. This yeilds a new formula that we can use to compute
interest that is compounded continuously.

Interest Compounded Continuously

This formula is for computing interest that computed and added to the balance of
an account every instant. This is not actually possible, but continuous compounding
is well-defined nevertheless as the upper bound of “regular” compound interest.
Notice that we have the same variables from our compound interest formula, except
the value in parenthesis has been replaced with e.

This formula can also be used for exponential growth and exponential decay. The
function of e is often called the exponential function because of its unique
properties. We must remember that e is a constant so it is still in exponential
function form.

Let’s do an example of interest compounded continuously. $1,000 dollars is deposited
at 14% per year, compounded continuously. Find the balance after 8 years.

First let’s define our variables. P = 1,000, r = 0.14, and t = 8,
so


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