 Welcome back to our lecture series, Math 1050, College of Oxford for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In our previous video, we had talked a lot about some financial math problems, particularly related to compounded interest and how these naturally lead to questions about what we're now gonna call exponential functions. So what is an exponential function exactly? Well, we are looking at functions that can be basic functions of the form F of X equals C times A to the X, where A is gonna be some positive, some positive number, real number, but it's not equal to one. I'll explain why we wanna throw out one in just a second. This is commonly referred to, the one here is commonly referred to as the base of this exponential function. We're taking functions of the form C times A to the X right here. Where A is this positive number, C is whatever we want, probably just not zero, because if you times it by zero, it would squish everything, right? In particular, an exponential function is a function where the exponent actually is the variable. This is in contrast to the power functions, which we had talked about previously in this series, power functions, their functions of the form Y equals, some C times X to the A. This would be something like X squared, X cubed, those type of things, right? Where in that situation, the base was the variable and the exponent was a constant. We're now trying to consider situations where the variable is now the exponent and the base is a constant, okay? Now some things to be aware about when you have exponential functions. First of all, if you take A and you raise it to the zero with power, that's gonna give you a one. And so when you plug in zero to our function f of zero, you're gonna get C times A to the zero, which is C of one, which is called, that's just the number C. So this coefficient C is actually the initial value of the function. It's the bi-intercept. And so that's an important thing to remember, assuming we don't do any transformations to our exponential function here. This C right here gives us the Y-intercept of it. That's gonna be an important value. When we talked about, for example, as we talked about compounded integers previously, right? The amount was equal to P times one plus R over N to the NT power. This is an exponential function with respect to the variable T where our base was equal to one plus R over N to the NT power. And we considered what happened as N went to infinity. We actually ended up getting the formula A equals P to the RT, right? Where again, we have this exponential function. We're working base E in this situation. The reason I bring this up is that I want to mention, coming to this formula right here, this P right here is the initial value. It's the principle, the initial investment. We would put that there as we did those financial problems. That's because that's what happens with exponential functions. This vertical stretch that's affecting the graph actually gives us the Y-intercept. Let me take that off the screen now. Another important observation here to look at is, if you were to compare the ratio of consecutive terms in the domain. So you take F of X plus one divided by F of X. Well, by the formula, you're gonna get C, C times A to the X plus one over C times A to the X for which the C's would cancel and you're left with just A to the X plus one over A to the X there, which by X one of rules, the top would factor as A to the X times A for which those then cancel leaving you just an A. So in particular, the base is gonna give us the rate of change of this function. Like how quickly it grows will depend upon this factor of A. If you have a big base, the function will grow faster. If you have a smaller base, it will grow slower, right? Let me give you some examples. I wanna start graphing these things right here. So for our first example, consider we wanna graph the function F of X. Oh, let's not put it there. Let me put it here on the screen. So let's say we wanna graph the function F of X. Oh boy, type over there. F of X is equal to three to the X. There you go, that's not horrible pinmanship whatsoever. Anywho, if we wanna graph that function, one idea is to search this graphing point. So if we did something like zero comma what, you put zero into the function, you're gonna get three to the zero, which is actually the same thing as the number one. So that gives us a first point on the graph. Like so, if we do another point, if you plug in X equals one, then the Y coordinate would be three to the first, which is the same thing as three, which you get that point right there, label. If we do the next one, we take two, right? The next one should be three squared, which is a nine. Let me put that back there. And actually, in second thought, let's actually leave it in this exponential notation. We have three to the first, and then one here was three to the zero. And I'm gonna have to zoom out a little bit so you can see all these points, you get a nine. If we did one more point, three comma three, three comma three cubed. Let's label it. Now we get that point right here. And so you can see, I'm gonna turn off these labels, you can see that on the right-hand side of this function, it grows really rapidly. As you take one step to the right, one, two, three steps to the right, it gets big, big, big, big. And so you can see kind of how the growth of that function is gonna work. If I were to try to connect the dots on this thing, we would get something like this where the steepness is really, really steep, right? You get a curve that looks something like that, all right? Well, what if we go the other way around? It turns out that we can actually allow for any exponent, right? We don't just have to take positive exponents, they don't just have to be whole numbers. We could take something like, what if we did one half? 0.5, we'd be taking three to the 0.5 power, which you can see matches up with my picture right there. Taking three to the 1.5 power is actually the same thing as taking the square root of three. Our exponents don't have to be whole numbers, they can be fractions, they can even be irrational. We could take pi comma three to the pi if we wanted to. And you see you're gonna get a picture up there at the top. Again, if we continued in the trajectory that we were drawing, that would be getting us the point right there as well. But what about negatives, right? What if I take something like three comma three to the negative one? One thing to remember about negative exponents. Oh, sorry, I need to take my x one should be negative one. What we see about negative exponents, I'm gonna erase my picture real quick so I can zoom in. When you take negative exponents here, you get the exponents gonna be 0.333, right? What is that number? This number right here is none other than one third. And that's what happens when you take a negative exponent. A negative exponent means you take the reciprocal. So when you plug in negative one, the reciprocal would give you one third. How about the next one? The next one here, we would plot would be negative two comma three to the negative two power. On which case, what point is that? That wants to be, the y-cord wants to be 0.111. Now, if you're curious what that number means, that number is in fact the same thing as one ninth. Where did one ninth come from? One ninth is actually one over three squared. And so when you take negative x, once you have to remember that you're taking reciprocals. And so if we do this one more time, we would end up with negative three comma three to the negative three, which that in fact gives us as a y-cord and one over 27. And so let's kind of zoom out and see what our picture has right here. So what you can see is that on the right-hand side, it grows very, very rapidly, right? But on the left-hand side, it gets smaller and smaller, smaller. And so now I'm gonna finally just type in the function here on the screen. We get y equals three to the x, like so. You see this one curve, it grabs all of them. This is the graph of an exponential function. On the right-hand side, you see that it does grow very, very rapidly. And that's because as you take one step to the right, you triple your value. If we take another step to the right, as you go from one to two, in terms of x-square, you're gonna triple. So this jump from here to here was a factor of three. But then when we do it again, we triple again, so we get up to nine. And then the next step, you triple it again. And so one thing to notice about exponential functions is that if you look at its in behavior as x approaches infinity, this actually tells us that y, let's try that again, that y will approach infinity as well. But this is somewhat of an understatement. Yeah, it goes towards infinity, but it goes to infinity extremely rapidly, crazy, crazy fast, faster than anything else we've been talking about in this lecture series. But it's also important to look at the other way around. As you take more and more negative values, right? As x approaches negative infinity for this function right here, we see that the y-coordinate is actually gonna be approaching zero from above. That is this function has a horizontal asymptote at the x-axis, but it only applies to the left-hand side of the function. That as you approach the left-hand side, you get this horizontal asymptote. And this is the basic shape of your standard exponential function. So this is the function y equals three to the x. Okay, what if we wanna try different, different exponential functions? So I'm gonna get rid of these points in play here. So let's switch my base so that it becomes this parameter a, right? And let's also pick a color so it distinguishes a little bit. Let's stick with blue. That seems good. So some things I want you to know about this function, right? Without any transformations, this function will go through the point zero comma one. That's its y-intercept. We also are gonna go through the point one comma a. You see that right there. And so now as a gets bigger, as you take bigger and bigger bases, the graph is gonna get steeper and steeper and steeper, right? The bigger the base, the steeper this thing gets, okay? But as a gets smaller, it gets more shallow, more shallow, right? And then I wanna kinda show you something. What happens as the a value gets close to one, right? It's getting really flat right there. So right here, we have the base of 1.1. It's really, it's not very big. It's, you know, it's not a whole lot bigger than one, but you still have this exponential growth. If we move it over to one, what you see is actually a flat line. And maybe that's no surprise here because if you take one to any power, right? What is that equal to? It's equal to one. So that's always gonna give you one. So you see this constant. And this is why we wanna rule out the possibility that a equals one, because that doesn't behave like other exponential functions. You actually just see that it's a constant, which we don't want that. If we take a value that's less than one, we'll talk a little bit more about this in a second, but this also looks like an exponential function, but it looks like things are behaving a little bit differently. It actually is exploding on the left-hand side and it's decreasing on the right-hand side. So when it comes to exponential functions, there's actually two types of exponential functions we talk about. There are growth models where this function's always decreasing. The base will be greater than one. And then there's what we call a decay models that if your base is less than one, you actually see that the function is decreasing over its domain. Now, on the other hand, if I take a negative value, turn this off for a second. If I take a negative base, you'll notice that Desmos doesn't have anything. Where did the graph go, right? Why can't you take a to equal negative one? Well, this gets really problematic. Exponential functions do have to be, have a positive base for the following reason. Let's say that, hey, you know, just for fun, what if a was equal to negative one? Then you have your function y, we'll call it f of x. You have your function right here. Let me switch over back over here. It's a little bit easier to write when I'm not on Desmos, but if you had your function like a equals negative one, then f of x would equal a to the x, right? What's going on there? Well, not, you know, some things aren't such a big deal. Like if you take f of zero, great. That's just gonna equal negative one to the zero with power which is one. You can do like f of two. That's gonna be negative one squared, which is one. All that seems hunky-dory. But what happens when you do something like f of one-half? Well, this would mean you have to take the square root of negative one, which is actually an imaginary number. And so this is what you see here is that exponential functions will have imaginary outputs if the base is negative. And so in order to retain our usual convention that input and output have to both be real numbers, we have to restrict our domains. That is we have to restrict the a value to be positive, something other than one. If we go back over to our picture here, let's look at some other examples of exponential functions we can get here. So I'm gonna switch things back. Well, I'm just gonna get rid of these in general here. Let's consider the following. What happens if we wanna graph the function y equals one minus three to the x? What happened here? Well, in this situation, let's compare this to y equals three to the x, just as a baseline here. And I'll make this one be dashed so you can see what the original graph was. So when you compare these things, like a couple of things happen, it looked like it's reflected across the x-axis because the blue one's actually pointing downward now, right? You'll notice that there's a negative sign here. Oh, that's a transformation. That's reflection across the x-axis. Well, what does this plus one do? Oh, okay, plus one would suggest that you've moved the graph up by one, okay? So we reflected it across the x-axis. I guess it should go the other way around. We reflected across the x-axis and then we moved it up by one. So the two transformations give us this graph right here. Transformations can be very, very helpful. What if we take the function one-third is my base? Like so, and so you take the exponential function whose base is one-third. Well, when you compare this, the blue graph, this right here is the function one-third to the x. The yellow graph is the function e to the x. They look like they're mirror images of each other, right? So the different base actually can create reflection across the y-axis and the reason for that is the following. One-third is the same thing as three to the negative one. And if you raise that to the x-power, putting the exponents together, you get three, three to the negative x. For which when one looks at something like that, you're like, oh, okay, the negative sign there is a reflection across the y-axis. That's a transformation we've seen before. So the decay model is actually just the reflection of the growth model. So that's what I was saying earlier that if your base is greater than one, you'll be increasing. If your base is less than one, it'll be decreasing and they'll actually be reflections of each other across the y-axis. All right, let's look at another example here. Let's try the function three to the x minus one power. All right, what's going on here? Well, that would look like a shift down by negative one. That is not what I meant to type in. I actually want the exponent to be negative one, x minus one. There we go. So let's try this one right here. So the blue graph is the one we started off with who has the coordinate zero one. This graph looks like in fact that we moved everything over by one. And it should, it says shift to the right by one. But one thing that's kind of curious about exponential functions is what if I did the following? Why not just one? What if I said h here? Let's take a slider here. And so when you look at this graph, the blue versus the yellow line right there, as I make the shift to the right to get bigger, like so I'm shifting it to the right, but an optical illusion might be playing on you right now. As I shift it to the right, am I shifting it to the right or am I stretching it horizontally? It's kind of curious, but also am I doing that or am I stretching it vertically? Or maybe I'm compressing it vertically. Like right, and you're shifting it to the right. I mean, you can see that maybe, but it kind of looks like also some type of stretch is coming into play. Why does the horizontal shift look like a stretch or compression of some kind? Well, the idea is the following. If we think about a little bit about exponent rules again, if you take three to the x minus h, this is the same thing as three to the x over three to the h by exponent rules. And so that is you're taking three to the x and you're dividing it by three to the h. And that was what I meant earlier by, doesn't it look like we're vertically compressing it? I think I said stretching earlier. We're vertically compressing it, which actually was the same thing as like a horizontal stretch here, right? And so when it comes to exponential functions, you don't have to worry about horizontal shifts, which is great, because those are some of the most difficult things to worry about here. The horizontal transformations are always more difficult than the vertical ones. For exponential functions, we do not have to worry about horizontal shifting because that actually can be reflected with the vertical stretch or compression, all right? Another thing I wanna show you, what if we did some type of not horizontal shift? What if we did some type of horizontal compression or stretch or something, right? How does that affect the graph? So notice here, what's happening as you are horizontally, I'm stretching the graph right here, you're stretching it out. This looks like an exponential just of a different base, right? As we adjust this thing around and think about the exponent rules one more time. By exponent rules, this is the same thing as three to the one over b, like so, raised to the x power. So it's like, okay, yeah, I can see that now. A horizontal stretch or compression will change the base of the exponential, but it still is an exponential function. And so this is one of my favorite things about exponential functions. When it comes to exponential functions, we never have to worry about horizontal transformations. You don't have to worry about horizontal reflections because that just changes the base. You don't have to worry about horizontal stretching or compressing because that just changes the base. And you don't have to worry about horizontal shifting because that's the same thing as a vertical stretch or compression. So when it comes to graphing a exponential function, there are only two things you have to worry about. Well, three things I really should say. The most general exponential function will look like the following. You have some capital C times a to the x plus k. So that's the only thing that you have to worry about here for which we're gonna set the base equal to three. I'm gonna turn this graph off right here. So as you change the base, that does affect how steep it is and such, right? So we'll just set it equal to three. So once you know the base, what does k do? k affects the horizontal shifting, right? It moves it up and down. Now, one thing that's important to remember is that this graph has a horizontal asymptote that we can see right here. As you shift of exponential function that also moves the horizontal asymptote up and down. So an untransformed, a non-transformed exponential, it's asymptotes the x-axis, this moves up and down, right? As you change the base, that doesn't affect the horizontal asymptote whatsoever. Okay, again, let's put it back to three. We'll put my shift back down to zero. What does C do? Well, as I change the x-intercept, as you change the x-intercept here, as you stretch it, that lifts it higher, right? A vertical stretch, you can do a vertical compression, right? Eventually, once you get negative, it reflects it across the axis there and stretches it again. And then you can combine this with stretching and shifting, right? And so you get all these different types of horizontal, all these different types of exponential functions, right? What make a little screensaver for us right here. And this shows us the gamut of what you can get with an exponential function. And so the graphs of them are pretty nice. We just have to worry about this general form. Let me write this on the screen before we end this video. So the most general form for an exponential function, if we do all the possible transformations, you'll get f of x equals C times a to the x plus k. That's our standard form for an exponential function.