Initial value & common ratio of exponential functions | High School Math | Khan Academy

So let's think about a function. I'll just give an example: let's say h of n is equal to ( \frac{1}{14} \times 2^n ).

So first of all, you might notice something interesting here: we have the variable, the input into our function, it's in the exponent. A function like this is called an exponential function. So this is an exponential, exponential, exponential function. That's because the variable, the input into our function, is sitting in its definition of what is the output of that function going to be; the input is in the exponent.

I could write another exponential function. I could write F of, let's say, the input is the variable T, is equal to ( 5 \times 3^T ). Once again, this is an exponential function.

Now, there are a couple of interesting things to think about with an exponential function. In fact, we'll explore many, many, many of them, but I'll get a little used to the terminology.

So one thing that you might see is the notion of an initial value. Initial, initial value. This is essentially the value of the function when the input is zero. So in these cases, the initial value for the function H is going to be H of 0. When we evaluate that, that's going to be ( \frac{1}{4} \times 2^0 ). Well, ( 2^0 ) power is just 1, so it's equal to ( \frac{1}{4} ).

So the initial value, at least in this case, seemed to just be that number that sits out here. We have the initial value times some number to this exponent, and we'll come up with the name for this number as well. But let's see if this was true over here for f of T.

So if we look at its initial value, F of 0 is going to be ( 5 \times 3^0 ), and same thing again: ( 3^0 ) is just 1; ( 5 \times 1 ) is just 5. So the initial value is once again that.

If you have exponential functions of this form, it makes sense. Your initial value, well, if you put zero in for the exponent, then the number raised to the exponent is just going to be one, and you're going to be left with that thing that you're multiplying by. Hopefully, that makes sense, but since you're looking at it, hopefully it does make a little bit.

Now, you might be saying, well, what do we call this number? What do we call that number there or that number there? And that's called the common ratio. The common, common ratio.

In my brain, we say, well, why is it called a common ratio? Well, if you thought about integer inputs into this, especially sequential integer inputs into it, you would see a pattern.

For example, h of, let me do this in that green color, h of 0 is equal to, we already established, ( \frac{1}{4} ). Now, what is h of 1 going to be equal to? It's going to be ( \frac{1}{4} \times 2^1 ), so it's going to be ( \frac{1}{4} \times 2 ).

What is h of 2 going to be equal to? Well, it's going to be ( \frac{1}{4} \times 2^2 ), so it's going to be ( \frac{1}{4} \times 2 \times 2 ), or we could just view this as this is going to be ( 2 \times h ) of 1.

Actually, I should have done this when I wrote this one out, but we could write as ( 2 \times h ) of zero. So notice if we were to take the ratio between h of 2 and h of 1, it would be two. If we would take the ratio between h of 1 and h of 0, it would be two.

That is the common ratio between successive whole number inputs into our function. So h of, I could say h of ( n + 1 ) over h of n is going to be equal to, actually, I can work it out mathematically:

[
\frac{\frac{1}{4} \times 2^{n + 1}}{\frac{1}{4} \times 2^n}
]

That cancels: ( 2^{n + 1} ) divided by ( 2^n ) is just going to be equal to two. That is your common ratio.

So for the function H, for the function f, our common ratio is three.

If we were to go the other way around, if someone said, "Hey, I have some function whose initial value," so let's say I have some function, I'll do this in a new color. I have some function G, and we know that its initial, initial value is five.

If someone were to say its common ratio, its common ratio is six. What would this exponential function look like? And they're telling you this is an exponential function.

Well, G of, let's say x, is the input, is going to be equal to our initial value, which is five. That's not a negative sign there; our initial value is five.

I'll write equals to make that clear, and then times our common ratio to the ( x ) power. So once again, initial value right over there, that's the five, and then our common ratio is the six right over there.

So hopefully that gets you a little bit familiar with some of the parts of an exponential function and why they are called what they are called.