Identifying graph for exponential

All right, we are asked to choose the graph of the function, and the function is f of x equal to 2 * 3^x. We have three choices here, so pause this video and see if you can determine which of these three graphs actually is the graph of f of x.

All right, now let's work through this together. Whenever I have a function like this, which is an exponential function because I'm taking some number and I'm multiplying it by some other number to some power, that tells me that I'm dealing with an exponential. I like to think about two things: what happens when x equals 0? What is the value of our function?

Well, when you just look at this function, this would be 2 * 3^0, which is equal to 3^0, which is 1. It's equal to 2. So one way to think about it in the graph of y is equal to f of x; when x is equal to 0, y is equal to 2. Another way to think about it is this value in an exponential function is sometimes called the initial value. If we were thinking of the x-axis instead of the x-axis, we would be thinking about the time axis or the t-axis. That's why it's sometimes called the initial value.

But the y-intercept is going to be described by that when you have a function of this form. You saw it right over there: f of 0, 3^0 is 1; you're just left with the 2. So which of these have a y-intercept of 2? Well, here the y-intercept looks like 1, here the y-intercept looks like 3, and here the y-intercept is 2.

So just through elimination through that alone, we can feel pretty good that this third graph is probably the choice. But let's keep analyzing it to feel even better about it and so that we have the skills for really any exponential function that we might run into.

Well, the other thing to realize is this number 3 is often referred to as a common ratio, and that's because every time you increase x by one, you're going to be taking 3 to a one higher power, or you're essentially going to be multiplying by 3 again. So, for example, f of 1 is going to be equal to 2 * 3^1, which is equal to 2 * 3, or 6.

So from f of 0 to f of 1, you essentially have to multiply by 3, and you keep multiplying by 3. f of 2: f of 2, you're going to multiply by 3 again. It's going to be 2 * 3^2, which is equal to 18. So once again, when I increase my x by one, I'm multiplying the value of my function by 3.

Let's just see which of these do this. This one, we said it has the wrong y-intercept. But as we go from x equal to 0 to x equal to 1, we are going from 1 to 3, and then we are going from 3 till it looks like we're close, pretty close to 9. So it does look like this does have a common ratio of 3; it just has a different y-intercept than the function we care about.

This looks like the graph f of x is equal to just 1 * 3^x. Here we're starting at 3, and then when x equals 1, it looks like we are doubling every time x increases by one. So this looks like the graph of y is equal to, I have my what we could call our initial value or our y-intercept, 3, and if we're doubling every time we increase by 1, 3 * 2^x that's this graph here.

As I said, this first graph looks like y is equal to 1 * 3^x; we are tripling every time, 1 * 3^x, or we could just say y is equal to 3^x. Now this one here better work because we already picked it as our solution, so let's see if that's actually the case.

So as we increase by one, we should multiply by 3. So 2 * 3 is indeed 6, and then when you increase by another one, we should go to 18; that's kind of off the charts here, but it does seem reasonable to see that we are multiplying by 3 every time.

You could also go the other way; if you're going down by one, you should be dividing by 3. So 2 divided by 3, this does look pretty close to 2/3. So we should feel very good about our third choice.