## __Exponents and Exponential Functions__

## What does a function look like?

This is a very simple function.

The function reads as

The function reads as

*"y equals 2 to the x power"*__Exponent Refresher:__

That equation tells you how to multiply x to get y.

If we want to find y when x=3, we can pretty quickly find that y=3*3=9. But, this is actually what's known as a "power

Take a closer look at x2 . This means x squared or x to the second power. What does it mean?

We have two parts here:

1) An exponent, which is the number 2.

2) A base, which is the variable x.

With exponential functions, the variable will actually be the exponent, with a constant as the base.

If we want to find y when x=3, we can pretty quickly find that y=3*3=9. But, this is actually what's known as a "power

__function"__. In fact, it's just a polynomial, and not an exponential function at all.Take a closer look at x2 . This means x squared or x to the second power. What does it mean?

We have two parts here:

1) An exponent, which is the number 2.

2) A base, which is the variable x.

With exponential functions, the variable will actually be the exponent, with a constant as the base.

__Exponential Functions:__

The eqation is y equals 2 raised to the x power. This sort of equations represents what we call " exponential growth" or "exonential decay". In the picture shows examples of other functions.

The general exponential function looks like this: , where the base b is any positive constant. The base b could be 1, but remember that 1 to any power is just 1, so it's a particularly boring exponential function.

The general exponential function looks like this: , where the base b is any positive constant. The base b could be 1, but remember that 1 to any power is just 1, so it's a particularly boring exponential function.

__Examples:__

Solve for x: 4= 2x

This one is actually pretty simple, so let's just think it through:

The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two:

This one is actually pretty simple, so let's just think it through:

The problem says we have to multiply x number of two's together to get four. Well, everyone knows that 2*2=4, so the answer is two:

__What about a word problem example?__

We can use a formula for exponential growth to model the population of a bacteria. Let's say the bacteria population is defined by B (T)= 100* 1.12t where B is the total population and t represents time in hours. While that may look complicated, it really tells us that the bacteria grows by 12 percent every hour. Every time another hour goes by, t goes up by 1, so we have to multiply the population times 1.12 again. The 100 simply sets the initial population at time t=0.

So, how much bacteria remains after 4 hours?

What do we know? We have the formula and the fact that t=4.

Replace t with 4 hours in the formula above and simplify.