Analyzing graphs of exponential functions | High School Math | Khan Academy

So we have the graph of an exponential function here, and the function is m of x. What I want to do is figure out what m of 6 is going to be equal to. And like always, pause the video and see if you can work it out.

Well, as I mentioned, this is an exponential function, so m is going to take the form. Let me write it this way: m of x is going to take the form a times r to the x power, where a is our initial value and r is our common ratio. Well, the initial value is pretty straightforward; it's just going to be what m of 0 is. So a is going to be equal to m of 0.

We can just look at this graph. When x is equal to 0, the function is equal to 9. So it's equal to 9. Now we need to figure out our common ratio. Let me set up a little bit of a table here just to help us with this. So let me draw some straight lines.

This is x and m of x. We already know that when x is 0, m of x is equal to 9. We also know when x is—let's see—when x is 1, when x is 1, m of x is 3. So when we increase our x by one, what happened to our m of x? Well, what did we have to multiply it by? Well, to go from 9 to 3, you multiplied by one-third.

So that's going to be our common ratio. In fact, if we wanted to care what m of 2 is going to be, we would multiply by one-third again, and m of 2 should be equal to 1. We see that right over here; m of 2 is indeed equal to 1. So our common ratio, our common ratio right over here is equal to one-third.

Now, m of x we can write as m of x is going to be equal to our initial value a, which we already figured out as a. A is equal to 9, so it's going to be 9 times our common ratio, times our common ratio one-third to the x power.

I was able to figure out the formula for our definition for f of x, but that's not what I wanted; I just wanted to figure out what m of 6 is going to be. So we can write down that m of 6 is going to be 9 times one over three to the sixth power.

Let's see, that is going to be equal to—that's the same thing as 9 times—well, 1 to the sixth is just 1. It's going to be 1 to the sixth, which is just 1 over three to the sixth power. Now what is three to the sixth power? In fact, I could even simplify this a little bit more. I could recognize that 9 is 3 squared, so I could say this is going to be 3 squared over 3 to the sixth.

3 squared over 3 to the sixth, and then I could tackle this a couple of ways. I could just divide the numerator and the denominator by 3 squared, in which case I would get 1 over 3 to the fourth power. Or another way to think about it, this would be the same thing as 3 to the 2 minus 6 power, which is the same thing as 3 to the negative 4 power, which, of course, is the same thing as 1 over 3 to the fourth.

So what's three to the fourth? So three squared is nine, three to the third is 27, three to the fourth is 81. So this is going to be equal to one over eighty-one. m of six is equal to one over eighty-one.

We could also have done that if we kept going by our table. m of 3 multiplied by one-third is going to be one-third. m of 4 multiplied by one-third again is going to be one-ninth. And we could say m of 5 is going to be, and m of 6 is going to be one-eighty-first.