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Analyzing graphs of exponential functions

Given the graph of an exponential function, Sal finds the formula of the function and a value that is outside the graph.

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Video transcript

- [Voiceover] So we have the graph of an exponential function here, and the function is m of x. And what I want to do is figure out what is m of six 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 xth power, where a is our initial power, and r is our common ratio. Well the initial value is pretty straightforward. It's just going to be what m of zero is. So a is going to be equal to m of zero, and we can just look at this graph. When x is equal to zero, the function is equal to nine. So it's equal to nine, and now we need to figure out our common ratio. So let me set up a little bit of a table here, just to help us with this. So let me draw some straight lines. And so this is x and m of x. We already know that when x is zero, m of x is equal to nine. We also know when x is, let's see. When x is one, when x is one, m of x is three. M of x is three. 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 nine to three, you multiplied by 1/3. So that's going to be our common ratio. And in fact, if we wanted to care what m of two is going to be, we would multiply by 1/3 again. And m of two should be equal to one, and we see that right over here. M of two is, indeed, equal to one. So our common ratio, our common ratio right over here is equal to 1/3. So m of x, we can write it as, m of x is going to be equal to our initial value, a, which we already figured it out, as a is, a is equal to nine, so it's going to be nine, times our common ratio, times our common ratio, 1/3 to the xth power. So I was able to figure out the formula for our definition for m of x, but that's not what I wanted. I just wanted to figure out what m of six is going to be. So we can write down that m of six, m of six is going to be nine times one over three to the sixth power. Let's see, that is going to be equal to, that's the same thing as nine times, well one to the six is just one. That's just going to be one to the six, which is just one, over three to the sixth power. Now what is three to the sixth power? In fact, I can even simplify this a little bit more. I can recognize that nine is three squared, so I could say this is going to be three squared over three to the sixth. Three squared over three to the sixth, and then, I could tackle this a couple of ways. I could just divide the numerator into the denominator by three squared, in which case, I would get one over three to the fourth power. Or another way to think about it, this should be the same thing as three to the two minus sixth power, which is the same thing as three to the negative four power, which is, of course, the same thing as one over three 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 81. M of six is equal to one over 81. We could also, we could've done that if we kept going by our table. M of three, multiply it by 1/3, is going to be 1/3. M of four, multiply by 1/3 again, is going to be 1/9. Then we could, m of five is going to be 1/27, and m of six is going to be 181st.