 This video will talk about exponential functions in an introduction. If we look at our definitions, exponential function can be written in f of x is equal to a times b to the x. Where b it says down here, it's our base on the exponent of x, which is an unknown. a and b are both real numbers. a can't be zero because zero times anything would just be a function of zero, and that's not going to be an exponential. And b has to be greater than zero and it can't be equal to one. So we want to look at this f of x is equal to three times two to the x. And over here on my calculator, I actually have put that in and I'm looking at the table. I want to look for some patterns. So let's look at the x. We can see that the pattern is that it's increasing by one. And then we want to look at the y's. And we can see the pattern happening there. We might be doubling because if you double seven point five, you get to one point five and double that, you get three, double that, you get six. So it looks like we are doubling or we could also say that we're multiplying by two to get from one y to the next. Let's look at another example. Now we want to look at this function and see if we can see the patterns that happen here. So we need to clear this one out and put in six and then in parentheses. I'm just going to put point five for that one half. Close the parentheses, clear it x and then go look at my table again. And this time again, let's look at x and see what's happening. X is again increasing by one. And if I look at the y, now my y values are getting smaller and it looks like they're getting smaller by half. They're decreasing by one half. Let's see if we can make some sense then out of those patterns that we just looked at. This means that the coefficient of A, if we were to go back and look at those, the coefficient, let's look at this one over here that we already have in here, the coefficient was six. And if I look back at my table, zero six was a point on my graph. So A tells us the y-intercept. When x is zero, we know what the y-intercept is. And B tells us how the function changes. Now this particular one, B was one half. So that would be this case down here in red, where B is greater than zero but less than one. Our function was decreasing. And in the other case, when we had three times two, remember we were multiplying by two and our numbers kept getting bigger so it increases when B is greater than one. So let's find an exponential function now and go the other way. Let's look at a table and see if we can find the exponential function. So let's answer these questions to see if that will help us. How is the input of x changing? Well, it's changing by one, increase by one. How are the y values changing? Well, it looks like from 20 to 60 I might have multiplied by three and 60 times three would be 180. So we're multiplying by three. How much should we start with? That's when x equals zero and when x equals zero we started with 20. So that means that my 20 is my A and the three is my B. As long as x is increasing by one, I should just be able to write this function as f of x is equal to A which is 20 times B which is three to the x. Our last example here then we're going to look at what happens when x doesn't change by one. If you look at this we're adding three each time to get from one x to the next. So x is increasing by three and the y's are decreasing. We are going from 3200 to 800. It looks like we divided by four and 800 divided by four would be 200. And 200 divided by four would be 50 and so on. That's a divide. We divide by one fourth. We could say that we divide by one fourth or the same thing is also multiplying by one over four. Same thing as dividing by four. Now how much should we start with? Remember x is zero so when x is zero we say that we started with 3200. So 3200 is our A. One fourth is also our B. But we've got a little bit of problem here because if I were to put that in my calculator and then double check my table to see if I did it right. Right now it feels like I should be able to say 3200 is my A times 0.25 which is one fourth. But if I do care at x let's set up my table to do what I wanted to do. So second window and I wanted to start at zero and x is changing by three in that table so I want my table to change by three to confirm. And I do start at 3200 but the way I had it written when it's three it should be 50 not 800. So it's really 3200 and it is 0.25 but the exponent it doesn't happen every one. That would be x but since it happens every three it's every third or x divided by three. So if we come back in here and fix our exponent and I need to put it in parentheses since it's a fraction. Now a parentheses and then x divided by three. Close the parentheses. Now if I look at that table that's going every three 3200 800 250 just like I expected it to be. So when it doesn't happen every one then you're going to have to divide by how often it does happen.