Subscribe! http://www.freemathvideos.com Welcome, ladies and gentlemen. So what I'd like to do is show you how to graph exponential equations. And we're going to graph exponential equations that are in the form of y equals b to the x, which are the first three, and then also are down here and down there. And also equations that are in the form of y equals a times b to the x. Now it's important, again, to understand that b represents the base of the exponent. x represents the power of the exponent. And a is going to be the coefficient. So you can see here, let's see. 3 is a. b Is 1/2. And then we have negative x here as our power. Now it's very important to be able to distinguish them because a lot of time students get confused. But just remember a is not being raised to that power.
Now if we don't have an a, remember we can always take a look at saying my a going to be 1 in front of that. And also I just pointed out the exponential equation e because e is a number. It's not representing a variable. We'll talk more about that later on.
Now you notice I don't have graph paper, right? I'm just going to be sketching the graphs. So I'm not going to be very exact. But basically what I tell my students obviously using graphing technology, a graphing calculator, or your phone or a computer, you can understand exactly what the graph looks like. And if you want to spend more time, you could create a table, which we're going to do for the first example just to kind of get an idea of how to always take a look at the graph to get it exact. But for this video, I'm not going to be concerned with finding the exact value of what the graph is going to look like. I'm just going to kind of get a rough sketch based on the transformations.
All right. So the main thing, though, I want to look at here is I'm going to graph this third one and kind of draw some understanding basically of what the exponential graph is going to look like. So one thing I always tell my students is, if we want to graph something, the best and easiest way to always graph-- no matter what function we're dealing with-- is to create a table. And I'm just going to this for the first example just so you can kind of get an idea. So let's just kind of pick some numbers. And I'm not going to go crazy.
Let's just do this. This, I think, hopefully will give you a good idea of what we're doing. Now these are going to be the x values I arbitrarily chose. You can choose any values. But these values are pretty simple for me to be able to plug into x to find y. So if I did y equals 3 to the negative 1, that equals 1 over 3 to the 1, which equals 1/3, which is like 0.333. Right? And I'm going to use the decimal approximation because I want to graph it. And sometimes graphing fractions gets a little confusing for students. So I think if we look at them as a decimal, it'll be a little bit easier. Y equals 3 to the 0. Any number raised to the 0 power is going to equal to 1. y equals 3 to the first power is just equal to 3. And y equals 3 squared is equal to 9. OK?
Now if I simply just wanted to plot these points, what I'd have is at negative 1-- so 1, 2, 3, 4, 5, 6, 7, 8, 9. So at negative 1, I'm only up 1/3. At 0, I'm at 1. At one, I'm up 3. 1, 2. 1, 2, 3. And at 2, I'm up 9. All the way up there. OK? So what you can see is, as we're going to keep on going, this graph is going to keep on getting larger and larger. And if I would have chosen negative 2, what that would have given me is 1/9. So at negative 2, I would have gotten even smaller and smaller.
And what we'll notice is, as you keep on going farther down again, you can use a graphing technology for this. What you'll do is you're going to get closer and closer to 0. But we're never actually going to approach 0. So what that creates is what we call an asymptote. The graph is approaching 0. But it's not actually ever going to reach 0. So that creates an asymptote, which we represent as a dashed line.
Now you can see on the other side, as we're going to positive infinity, what this graph is going to do is ever increase. So if I just wanted to kind of sketch this graph, it's going to look something like that. OK? Now that's for y equals 3 to the x, which I guess I probably could have just graphed right here. Because I wanted to do something else over there.
OK. So that's that graph. And I'm going to graph the asymptote because the asymptote is a part of the graph. And that's one thing that a lot of students kind of forget, they don't graph the asymptotes. And what happens is that starts to confuse them once we start talking about domain and range which you are going to do for all these problems as well.
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