What is Euler's Number?

What is Euler's Number? - How it Works - Videos 1 & 2

Exponential Functions - What are they?

Exponential Functions - What are they?

We have seen y = x², y = x³, y = x⁴, and, ... What happens when we switch the power or index and the variable. We get y = 2^x, y = 3^x, y = 4^x and, ...

Now another another name for power or index is exponent. We can extrapolate that to exponential function. We have two types of exponential functions. One is exponential growth, where a is greater than 1. And the other is exponential decay, where a is in between 0 and 1.

Theorem: Exponential Functions are One-to-One

Theorem: Exponential Functions are One-to-One:

The theorem states that for every input value there is one output. The only time the results are the same is when the inputs are the same.

Example

Example:

In this example we have y = 2^x. y is the result. 2 is the base. x is the exponent.

Let's make an XY table, and choose 3 as the first input value. 2⁰ is 1, so we have our first point (0, 1). The next point going right is (1, 2) so we double our result. The next point is (2, 4). Once again we doubled our result. The next point is (3, 8), and our result doubled again. Why does it double? Because are base is 2. If our base were 3, then it would triple.

Now let's go to the left of 0 or the negative numbers. Will the result be negative? Since the input value is an exponent, the negative value will flip the base to the numerator and switch the exponent to positive because of the rules of exponents.

Another way of thinking about this, 2^-1 is 1/2 so our point is (-1, 1/2). 1/2 is half of 1. The next point going to the left is (-2, 1/4) so we halved our result. The next point is (-3, 1/8), and our result is halved again. Each time we move to the left one unit, our result is halved. If our original number is positive, will the result be negative? No, a positive divided by a positive will always be a positive.

Which leads us to one more question. Will the result ever be 0? No. Although 1/512, 1/1024, .... 1/32,768 are very small, they are not quite zero. This tells us that we have a horizontal asymptote at y = 0 or the x-axis.

Compound Interest

Compound Interest:

One of the ways to use exponential functions is with money. In simple interest, the money increases based on the principle amount and interest rate and excludes any money gained from the interest.

With compound interest the money increases based on the principle amount and the interest rate and includes any money gained from the interest. So with compound interest your money will increased at a much faster rate.

Live Worksheet

Teacher - Edpuzzle Link