 This video is going to talk about exponential graphs. Now hopefully you have looked at this website that we have over here to the left-hand side, and you've played around with what A happens with A and B, and we're going to take a look at when A is greater than 0 and B is greater than 1. So let's move the A, and you can see that as A increases, if you look closely, it's moving up and down the y-axis as we increase or decrease that A. If we move the B, as it gets greater than 1, and it continues to increase and gets steeper, and as we make our B get smaller again, still greater than 1, it's going to be getting flatter. But the graph, when B is greater than 1 and A is greater than 0, is first of all, it's going to be increasing. If you look over at the graph, as x gets larger, y is also getting larger. So it's an increasing graph. The other thing that we notice about this graph is that it is always above the x-axis. So we can say that if A is greater than 0 and B is greater than 1, our graph will be above the x-axis. So what happens if A is less than 0, but we're still looking at B greater than 1? So let's make those A's less than 0. And now you can see that our graph is again moving up and down the y-axis, but now it looks like it's going to be decreasing. So it's a decreasing graph, B is greater than 1, A less than 0. We'll have a decreasing graph, and it's going to be below the x-axis. And that's because of the A being less than 0. Now if we look at our other case where we have B is between 0 and 1, and A is greater than 0, going back and looking at our graphic, now we can see that as we move A, it moves up and down the y-axis again, and that we can also see that the graph is decreasing as x gets larger, then y is going to be decreasing, and these graphs were above the x-axis. Okay, so if A is less than 0 and B is between 0 and 1, we can see that this graph is increasing, and we can also see that it was below the x-axis. And remember that's because again the A is less than 0. So let's see if we can make some generalizations about all of these graphs so that we would be able to graph and cohesion if they gave us one. So in general, we could say that if A is greater than 0 and B is between B is greater than 1, then we can say that our graph would be an increasing graph. And if B is between 0 and 1, then we would say our graph is decreasing. Now if B tells us increasing or decreasing, the A also tells us a couple of things. One thing it tells us is which side of the x-axis is this graph going to be on. If it's greater than 0, we know that it's going to be above the x-axis. But the other thing that A tells us that's a little bit harder to see in the graphic that we were looking at, it tells us what the y-intercept is. It will always be the y-intercept. So in general, we know that we have increasing and decreasing graphs. But if A is less than 0, we actually could just say what we know, everything is going to be the opposite of what we said. So if B is greater than 1, instead of being increasing, it will be decreasing. If B is between 0 and 1 and A is less than 0, then instead of being decreasing, it's going to become an increasing graph. So everything is the opposite. In general, we can just remember that B greater than 1 is increasing, B between 0 and 1 is decreasing. And if A is less than 0, then everything will be the opposite of what we think. That way, you'll have to remember one rule. All right, so if we sketch some graphs, we can do these by hand just by looking at the equation. So we look at this and we see that A is equal to 4. That means it's greater than 0. B is equal to 5. So again, A is greater than 0 and our B was greater than 1. So we should know that this is going to be a graph that crosses the y-axis at 4 and then it's going to have to increase through that point since B is greater than 1. So it'll increase through that point. Start low and then hit that point and continue up. Not a great sketch, but again, it's just a sketch. Okay, and our second example then over here, we have A is negative 10 and B is 3-4. So A this time is less than 0. So that tells us it's going to be below the x-axis and B is between 0 and 1. So that tells us that it's going to do the opposite of what we think. So let's sketch our graph. We have our graph here. Let's put our negative 10 down there as our intercept. So 0 negative 10 on the y-axis or our y-intercept. And I'm going to do these by 2. So 2, 4, 6, 8, negative 10, and there's my y-intercept. B is between 0 and 1, so it should... I would think it would be decreasing, but it's below the x-axis, so I know that it has to increase through that graph or through that point. So I'm going to start below my negative 10, come up through it and then continue on increasing. And I have an increasing graph. Let's look at some characteristics now of graphs. The x-intercept. Well, with an x-intercept, we can say that there are no x-intercepts. If you look at those graphs carefully, if we put like an increasing graph in here, B greater than 1, A greater than 0, we can see that this graph goes forever and ever left and right, but it never crosses the x-axis. If we take a look at a graph in our calculator here, 2 times 2 to the x. And then we do a standard window and we look at the graph. It looks very much like my sketch, but you can see that it never crosses the x-axis. It never gets below that x-axis. If I look in my calculator and I continue to go up and up and up and up, and I get into those, they're just getting closer and closer to 0, but it never becomes 0. So we have no x-intercepts on exponential graphs that are general. Let's talk about the y-intercept then. The y-intercept, remember that we said that that was the A and the point would be 0, A. In our equation, we could find the y-intercept as that constant A. So let's look at our graph. We said it was 2 times 2 and the 2 in front. That was my A and it was 0 too. We just saw in our table was the point that we had. Now let's think about the domain. How far to the left and how far to the right is this graph going to go? This is my sketch of my graph. And this graph is going to go forever, left and forever to the right. So we can say that the domain is going to be all reels. Let's think about the range values then. Remember that we don't have an x-intercept. So that means that we never quite get down to y equals 0, but it increases from above that x-axis. So we would say that y is greater than 0. Or if we put it in interval notation, we'd have to say 0 with the parentheses to infinity because we can't include the 0. Okay, the horizontal asymptote. Horizontal asymptote, remember, is asymptotes are imaginary lines that the graph tends toward but never crosses. Well, we've already discussed that this graph gets real real close to the y-axis but never crosses it. So our horizontal asymptote would be that y equals 0, or the x-axis. And then if we look at this particular graph, talking about increasing or decreasing, as b is greater than 1 in this case, so it's increasing. If b would be between 0 and 1, that's when the y-values are getting smaller as x-values get larger. So let's look at a final example. We want to look at the specific graph and we want to know if f of x is increasing or decreasing. And we can look over here as x is increasing, we can see that the y-values are also increasing. So this is an increasing function. Now, is this considered exponential growth or decay? Well, if we think about in real life situations, if something is getting bigger and bigger and bigger, it's a growth. If it's getting smaller and smaller and smaller, we call it a decay. So we would call this one an exponential growth. Now, if I use my graph to estimate for f of 1, this is an x in here, so I go to 1 on the x-value and go and see until I hit my graph and it looks like it's approximately across from the 9. So I'm going to say approximately 9. Now I'm going to use the graph to find x when f of x is equal to 1. Well, that's this y, or this 1 is a y-value. I'm going to change colors here. So I go to 1 on the y and go across until I hit my graph and it's kind of hard to tell exactly, but I'm going to say that it looks like x is about 1. It's approximate again. And it's equal to negative 1 now. So if vertical intercept, that's easy. That's right there on the graph. We can see that one, no problem. We know that that one is 0, 3. So now we need to write a possible equation for f of x. Well, remember that the y-intercept is our A. So A in this case is going to be equal to 3. So now we at least know that f of x is equal to 3 times something. We have to figure out what that something is. We know that b is greater than 1 because we're increasing here. f of x is equal to 1 when x was 9. That meant that it was 3 times 3 to the first. So I think the base is going to be 3 times 3, so we'll say 3 times 3 to the x. And that would be my function.