 Welcome to Episode 16. We're starting some new material today in Math 1050, College Algebra. I'm Dennis Allison in the Math Department here at Utah Valley State College. And we're starting a new chapter on exponential functions and logarithms. Today, we'll be talking about exponential functions. And next time in Episode 17, we'll look at logarithms. Let's go to our list of objectives for today. First of all, I'd like to review some fundamental properties of exponents that I think you're probably familiar with, that we just want to mention just before we get this started. Then we're going to look at some new fundamental graphs. These are fundamental graphs of exponential functions. We'll be making transformations of these exponential graphs. And then we'll look at one special exponential function called the so-called natural exponential function. OK, let's go to our list of review properties. OK, I'm wondering if we all remember these properties of exponents that we've covered in elementary and intermediate algebra before we look at exponential functions in this section. Who can tell me what is the product A to the m times A to the n? A to the m plus n. A to the m plus n, right. So when you're multiplying like bases, you add the exponents. What about A to the m over A to the n? A to the m minus n. A to the m minus n. By the way, we have a qualifier over here that A can't be 0. Why is that? Can't divide by 0. Yeah, because you see A to any power would be 0 and you can't divide by 0. So we, of course, have to make that one stipulation. What about A to the m quantity raised to the n power? What would that be? A to the m times n. A to the mn power. Yeah, m times n. Exactly. Very good. Finally, also, what is A to the 0 power? 1. A to the 0 power is 1 whenever A is not 0. A to the 0 power is 1. What is 0 to the 0 power? Just to ask you offhand. 1. No, actually, it's not defined. 0 to the 0 power has no meaning. Kind of like you can't divide by 0. You can't raise 0 to the 0 power. The reason for this rule, A to the 0 power being 1, is basically an extension of this very first rule. You see, if I were multiplying over here on the side, if I were multiplying A to the 3rd power times A to the 0, if I were using this first rule, I would add the exponents. And I would get A to the 3 plus 0 is 3. So what must A to the 0 be? Because A cubed times this is A cubed. That must be what? 1. That must be 1. So that's why we just define it to be 1, so that I can continue to use that rule. Then finally, our last expression is A to the negative n. And what is another way of expressing A to the negative n power? 1 over A to the n. 1 over A to the n, exactly. And the reason for that, let me just put that in the same space. You see, these are merely definitions that we've been given, that we've given in mathematics. If you multiply A to the 3rd times A to the negative 3, if I use this very first rule up here, I should add exponents. And I would get A to the 3 plus negative 3 is 0, which we said is 1. So that means A to the negative 3 must be the reciprocal of A cubed. So we define this to be 1 over A to the 3rd. So in general, A to the negative n is 1 over A to the n power. OK, those are some of the fundamental properties of exponents that we have seen in the past. And we'll be using those in this episode. OK, let's now look at how we find exponential values on a calculator. And let me bring my calculator in and set it up on the green screen. If we could focus in, we could zoom in on this calculator right here. Not everyone at home, maybe not all of you, or maybe perhaps comfortable with taking exponential values on a calculator. So I want to show you how I can do that right here. Suppose I want to take 3 squared. So if I turn my calculator on, I'll enter 3. And then I have this button over here with sort of a little, let me move that up so you can see it. It has this little carrot shape, this little inverted V shape. That's the exponential key. And if I push that, what it says is the next number I enter is the exponent. So this now says 3 squared, 3 to the 2nd power. And if I press Enter, then I get 9, 9. Let's do that one more time. Suppose I enter 12, and I want to raise it to the third power. So I'll raise it to the third power, and I get 17, 28. Now not every calculator has this symbol for their exponential key. And if we go to the next graphic, I'll show you what other calculators sometimes have. Let's go to the next graphic. Here we go. So you see in these two examples, we have 3 raised to the 2nd power. And then when I press Enter, that's the equal sign, then I got a 9. But on some calculators, you might have a key that says y to the x power. And then on that bottom line, we have 125 raised to the power of, and then I put 1 third in parentheses. Parentheses, 1 divided by 3. So that's 125 to the 1 third power equals. And a 1 third power is a cube root. So the cube root of 125 is 5. And so you can see what the more typical expression for that relationship is on the right-hand side. The cube root of 125 is 5. So you'll be using in this episode either that little inverted V key for exponents or y to the x. And I think there may be some calculators that have x to the y. But any of those should be able to give you an exponent. OK, now let's go to our fundamental exponential functions. And to do that, let's go to the next graphic. A function such as f of x equals 2 raised to the x power with a variable in the exponent is called an exponential function. And here we're asked to sketch the graph of the exponential function f of x equals 2 to the x. OK, let's go to the green screen and see how we would graph this. f of x equals 2 raised to the x power. Now you notice this time I have a constant in the base and I have a variable in the exponent. So that's why we call this an exponential function. And I've never graphed this before. So as we've seen in the past, I'm going to make a table plot points. But of course, that always gets to be rather long and tedious. I'm going to look for target points and find a quicker way to graph this. So let's set up our table over here on the side and we'll label our columns x and 2 to the x. Now suppose we substitute in 0, 1, 2, 3, or 4. The function values would be 2 to the 0, which is 1, as we just saw. What's 2 to the first power is 2. Yeah, 2 to the first power is just 2. What would be the function value for 2? 2 to the second power is 4. And 2 to the third power is 8. And 2 to the fourth power, 2 to the fourth power is what? 16. Now you notice what's happening is every time x increases by 1, 0, 1, 2, 3, 4, my function values double. 1, 2, 4, 8, 16. So I bet the next value would be 32 if I were to substitute in a 5. So the values on the right are said to be increasing exponentially because they're doubling. The values on the left, the x values, are increasing linearly because they're increasing by 1 every time. Now what if I substituted in negative numbers like negative 1, negative 2, negative 3, negative 4, et cetera? Well if I substituted in negative 1, I get 2 to the negative 1 power, which is what, Ben? Wouldn't that be 1 half? 1 half. You remember that was the last property on our list. 2 to a negative power means 1 over 2 to the first power, and that's 1 half. If I substituted in negative 2, 2 to the negative 2 power, that's 1 over 2 to the second power, which is 1 fourth. What's 2 to the negative 3 power? 1 eighth. 1 eighth. And what's 2 to the negative 4 power? 1 sixth. 1 sixth. Now these numbers are decreasing linearly, negative 1, negative 2, negative 3, negative 4. And these numbers are decreasing exponentially because we're going down, these numbers are being cut in 1 half, 1 fourth, 1 eighth, 1 sixteenth. OK, now we have a lot of points there. Let's plot some, if not all of those, and see what our graph looks like. Let's label it out to 4 there, negative 4 here. OK, that's about 6 right there. I'll go down a bit negative. This is the y-axis, of course. So I'm going to be plotting some ordered pairs. My first ordered pair would be 0, 1. And if I plot 0, 1, I get a point right here. If I plot the point 1, 2, 1, 2, that would be right here. And then the next point I'd plot would be 2, what? 4. 2, 4, right, 2, 4, right there. And then 3, 8. Well, 8 would be, let's say, about there. And 4, 16, 5, 32, those are so big that they just go off the graph. So I think we can see that as the x's get bigger, this graph turns up very dramatically. And if I go to my negative values at negative 1, I would go up to 1 half right there. I could probably move that up just a shade. And then at negative 2, 1 fourth, so it's only half as tall. And at negative 3, it's 1 eighth, only half as high as it was there. And at negative 4, it's 1 sixteenth. And at this point, it's so close we really probably can't see the distinction between the point and the x-axis. So it looks to me like our graph looks like this. And this is a typical exponential function, the graph of a typical exponential function. Let me just write this over here. The graph of an exponential function. This function has an asymptote. You remember a couple episodes back we were talking about rational functions with vertical and horizontal asymptotes. This function has an asymptote. What type of asymptote does it have? Horizontal. It has a horizontal asymptote, the x-axis. The x-axis is a horizontal asymptote. However, it does not have a vertical asymptote. The reason is, even though this graph is climbing, there's no imaginary wall, no imaginary vertical line that it approaches but never reaches. This function just keeps spreading out to the right and getting higher and higher and higher. For example, if you were to go over to 10, which is beyond my graph, 2 to the 10th power would be over 1,000. You could check it on your calculator, it's 1,024. So at 10, it'll be at 1,000. And if you go beyond that, if you go further to the right, it'll be a million and then a billion and so forth. But there's never a point where it approaches a vertical line and can't cross it. Now, of course, in the future, I'll be graphing more exponential functions. And I want to look for a shortcut, so I don't have to make a table every time. So let's pick out some points here to be our target points. If we go back to this graph, the points I'm going to choose are the points 0, 1, and the point 1, 2, and the point negative 1, 1, 1, 1. Let me just list those over here. There's the point 0, 1, 1, 2, and negative 1, 1 half. Now, you notice it doesn't really matter what the base is. Say, if the base is 3 or 5 or 7, when I raise it to the 0 power, I always get a 1. So I always go through the point 0, 1, irregardless of whatever the base is. But if I go over 1 unit, then the distance that I go up is whatever the base number is. So if I go over 1, I get 2 to the first or 2. So whatever that base number is at 1, I go up the base. If that had been 3 to the x, I'd locate the point 1, 3. And if I go to the left one, to negative 1, can anyone tell me another way of explaining why that's 1 half? Steven? One over the base? It's one over the base, yes. Because whatever the base is, you have the base raised to the negative 1 power, so you get the reciprocal of the base. OK, now, with that shortcut, we can start catching these very quickly. Let's take a new exponential function. Suppose I want to graph the function 5 to the x power. So f of x equals 5 to the x power. We don't want to make that table, because that's much too tedious to have to list every time. Let's take some shortcuts like we've been doing all through the course. I'm going to plot only three points. First of all, at 0, I go up 1. Now, the reason for that is if I substitute in 0 for x, 5 to the 0 is 1. And if I substitute in a 1, I go up 5. There's 5 right there, because 5 to the first power is 5. And if I substitute in a negative 1, I go up 1 fifth. It's hard to locate that exactly, but we'll say that's 1 fifth. And now I know the general shape is that I have a horizontal asymptote for the negative x axis. And the graph turns, and it goes up very abruptly. As a matter of fact, the larger the base, the faster this curve turns up. And this is the graph of the exponential function 5 to the x. While we have this written on the screen, let me just mention that in later courses in business, in physics, in engineering, and in mathematics, you'll sometimes see this function written this way. F of x equals exp sub 5 of x. This means the same thing as 5 to the x. And you might say, well, why would anybody want to write it that way when 5 to the x is so much more compact? Well, you see, this is a function of x. And so if we want to give a name to the function, this does not identify x as a function of x. It just places the x up in the exponent. So the name of the function is called the exp function. That means the exponential function. And if you put a little 5 there, that means it's the function 5 to the x power. I'm not going to use this on exams, but I just want to mention this to you so that if in a later course you see a function called exp, you'll know that this is just a more familiar version of an exponential function. Let's take another function. Suppose I wanted to graph the function capital G of x equals 10 to the x power. Equals 10 to the x power. Can anyone tell me three points to plot? Three points to plot. Let's see, 2, 4, 6, 8, 10 is one unit higher than that. Three points to plot so that I can draw this graph. x equals 1. Now we need an ordered pair. Oh, so it'd be 1, 10. OK, 1, 10. If you go over 1, you go up the base. So 1, 10, yep. 0, 1. 0, 1 right here. That's the point 0, 1. And negative 1 and 1, 10. And negative 1, 1, 10. Now G is we can hardly squeeze that in there, but that's supposed to be the point 0 and 1, 10. Or if you prefer, 0, 0.1 in decimal form. So this tells me that the graph is going to turn very dramatically up. And whoops, we missed the point there. But you get the idea. It goes through that point. And over here, it levels off very, very quickly. So that it's almost indistinguishable from the x-axis. For example, at negative 2, how high is the graph at negative 2? 1, 100th. It's 1, 100th. It's 10 to the negative 2 power, or 1 over 100. And at negative 3, it's 1, 1,000. So you see, this thing collapses very quickly. And it goes up very dramatically. You know what this reminds me of is the graphs of the higher order polynomials, like x squared is a parabola, x cube was flatter in the middle. And it turns up more dramatically. x to the fourth is even flatter in the middle, turns up even more dramatically. So the same thing happens with the exponential functions when you increase the base rather than when you increase the exponent. Another way of expressing this function I'll just mention in passing is to say this is exp sub 10 of x, if you should ever see that written in another book. OK, let's look at another example of exponential functions. But I'm going to pick my base a little differently from those first few examples. Suppose we have g of x equals 1 half to the x power. Now, you see what makes this different is I'm picking a fraction smaller than 1, a positive fraction smaller than 1 in my base. When I go to graph this, I'm going to follow the same rule for my target points. We said that at 0, we'd go up one unit. So there's the point 0, 1. We said that if we go over 1, we'd go up the base. The base is a half. And if I go back one, I'll go up the reciprocal of the base. And what's the reciprocal of 1 half? Two. Two. So you see this time, I have a graph that's coming down. It's decreasing and approaching the positive x-axis as a horizontal asymptote. So I can summarize the differences between this graph and the previous graphs this way. If the base is a positive number, but smaller than 1, you get a decreasing function. And if the base is a number bigger than 1, you get an increasing function. We saw 2 to the x, 5 to the x, 10 to the x. Those were all increasing from left to right. But if you put in a base smaller than 1, it forces this point to be lower, and the reciprocal is higher, and this becomes a decreasing function. Now, I think a reasonable question is, why would anyone want to study exponential functions? And there are many applications for these. And some of these applications will study in this course. For example, population growth. Compound interest in banking accounts can grow exponentially. On the other hand, functions that are decreasing, like the one I've graphed right here, would be radioactive decay. If you have a radioactive substance with a certain half-life, it has less and less radioactive material in it over time. So this would represent the graph of the amount of radioactive material in that. If that's a little confusing to you, I'd say hang on in a later episode. We'll be talking about some of these very applications. And you can see how this is applied to banking, to business, to chemistry, and to physics. Susan? If that were a negative 1 half, would the graph be flipped over the y-axis? Well, you know, we'll only consider situations where the base is a positive number. Oh, you mean a negative in front? Yeah. Oh, OK. If you put a negative in front, that would flip the graph over. The y-axis or the x-axis? It would flip it over the x-axis. It would be inverted and it would be coming up from underneath. I thought Susan was going to ask, what happens if you put a negative number inside for the base like a negative 1 half quantity to the x-power? We won't be considering those functions because those are sort of intermittently defined. A negative 1 half power raised to a 1 half power, that would mean a square root of a negative of a half. We can get imaginary numbers in that case. So I think it's best to just leave those alone. Oh, Steven. What if we had a variable in both the base and the x-power? Oh, my goodness. OK. Well, now you're getting to some interesting situations there. That's another situation we don't cover in this course. But I suppose it's possible you could have a function like f of x equals x to the x-power. Let's just see. If you had x to the x-power, what would be f of 2? 4. It'd be 2 to the second power, 4. What would be f of 3? 27. It'd be 3 to the third power. So you'd be plugging both in the base and in the exponent. 3 to the third power would be 27. Although that's not what we will call an exponential function in this course because we want it constant in the base, I would suggest you try graphing that in your graphing calculator. And if you take a course in calculus, I bet you will see that function come up along the way. Let's go to the next graphic. And here are some facts that summarize what we said about exponential functions. First of all, the graph of f of x equals a to the x-power, if a is bigger than 1, will curve up on the right and approaches the x-axis on the left, the x-axis being a horizontal asymptote. Number two, the graph of f of x equals a to the x. If a is bigger than 0, but less than 1, it curves up on the left and approaches the x-axis on the right. And finally, one thing I haven't pointed out in these graphs, but I think you'll recognize, is number three, each function f of x equals a to the x, this is where a is different than 1, is a 1 to 1 function. You know, all of the graphs that I've just drawn here pass the horizontal line test. And what that means is that each of these has an inverse function. That's going to become a very fundamental idea in the next episode, the fact that these are 1 to 1, and therefore they have inverse functions. They're inverse functions we're going to call logarithms. The reason I excluded the case of a equals 1, if we come back to the Greenmore, let me just show you this situation. If we have f of x equals a to the x, and we know that a is bigger than 1, then I know the graph generally looks like that. And if we have f of x equals a to the x, now we said we're always choosing a to be bigger than 0. If it's less than 1, then the graph generally looks like this. And you know, in both these cases, our graphs pass the horizontal line test. See, if I draw horizontal lines through these things, they cross at only one point. Now, of course, if you draw a horizontal line down here, they don't even cross it at all. But to be a 1 to 1 function, it says that a horizontal line can cross it at most one time. But look at the one exception to that rule. What if I put in 1 in the base? You notice 1 is omitted here, and 1 is omitted up here. What does the graph of 1 to the x look like? 1. Yeah, 1 to the x is 1. 1 raised to any power is 1. Even to the 0 power, it's 1. And so when I graph that function, if this is 1, when I graph that function, it looks like this. f of x equals 1 to the x. Now, that's not a 1 to 1 function. It's a horizontal line. And so that's why I've ruled out the one case where a could be 1. OK, let's go to three functions in the next example, in the next graphic. And let's graph each one of these on the screen. There's f of x equals 5 to the x. Let me just write that one down here. f of x equals 5 to the x. We've actually drawn that one already, haven't we? Then there's the function g of t equals 1 third to the t. And now, here's the ringer. m of x equals 0.1 to the negative x power. Now that's the one that's truly different from anything we've graphed so far, or at least it looks different. OK, for 5 to the x, I think I chose this as an example while ago, but let's do it one more time. If I want to graph 5 to the x, I'll plot the point 0, 1, 1, 5, and negative 1, 1, 5. And my graph looks like this. Now you see how fast that is to be able to draw it that quickly. And if you remember, when I graphed 2 to the x, I made a table over here on the right-hand side. And oh my goodness, that was really tedious. Now things are speeding up with the target points. When I go to my second function, I'll have to label this the t-axis. And we don't need to mark off too many points for this one, I don't believe. The target points will be at 0, 1, at 1, 1 third, and at negative 1, 3. And here's the graph of g. That's the graph of g. OK, but now we'll see, are there any questions on either of those first two? OK, now we come to this last situation. I have 1 tenth to the negative x power. Now that negative and the fraction here all make this to be a little bit surreal, I think. So can anyone think of a way to simplify that? 1 over 10. OK, let's call this 1 over 10 to the negative x power. Now, can you think of a way to reduce that further? Becomes 10 to the x. Yeah, you see, this negative says you should invert the base. And so this is going to be 10 to the x power. So this is just another way of saying graph 10 to the x power. So if I graph 10 to the x, I think we can squeeze that one in right here. Let's say this is 1. And for the sake of argument, I'll just jump up here and say this is 10. So I don't try to mark off 10 units in that little space. So I'll go up one unit to the point 0, 1. I'll go over to 1, 10. And I'll go back to negative 1, 1 tenth. Wow, it's kind of hard to squeeze that one in there. And my graph turns very quickly and levels off. And we graphed 10 to the x a little while ago. But the purpose of this example is to show you that the author, and I guess just in a mysterious way, I can hide these functions and make them look different than what they really are. So it's just a matter of us recognizing the functions for what they can be simplified to and then graph them. Let me just take one more example like that, and I'll write that one on the green screen. Suppose you wanted to graph this function, let's say f of x equals 6 to the negative x power. Now, when you see the sixth there, you say, oh, yes, this graph is going to rise on the right-hand side, but there's a negative in the exponent. If I get rid of the negative, what would the new base be? 1 sixth. So actually, this function is going to be rising on the left. Let's just graph that. Here's 6, and I'll go up to the point 0, 1. And if I go over 1, I go up 1 sixth. And if I go to the left one, I go up 6. And this is the graph of 6 to the negative x. Yep, that's the graph of the function f. By the way, this function has another name. You could write it in the form exp sub 6 of negative x, because you see it's negative x, not an x up there. Another way you could write this function is to call it exp sub 1 sixth of x. So there are lots of ways of naming these functions. 6 to the negative x, 1 sixth to the x, are using this exp notation. And it may seem funny that a function can have different names, but you have different names. I have different names. Some people call me Dennis. Some people say Mr. Allison, whatever. So you go by different names, and people know who they're referring to. So functions go by different names, and we should recognize them by these various monikers. OK, let's go to the next example of three new functions. Let's see, we just did that one. Let's go to the next graphic. OK, these are transformations of fundamental exponential functions. f of x equals 4 to the x plus 1, and then two others. Let's take f of x equals 4 to the x plus 1 and graph it, and we'll come back to the other two. OK, so f of x equals 4 to the x plus 1. By the way, I should remind everyone that surely by now you know that all of these examples are worked out on the website. And so if you just go to the episode 16 web page, you'll find these examples and the graphs shown. So if you're hurriedly trying to copy down everything I'm writing on the green screen, it's probably hard to keep up with, but you can see this written out fully on the web pages. OK, this is the function 4 to the x with a transformation. Can anyone tell me what the transformation is for this graph? Shift it up one. Shift it up one, right. And that's the only change that's being made. So when I draw the graph, I'm going to have to locate a new origin. That's the name we've been giving. You better move y up a little bit higher there. And let's see, you know we said that the x-axis was a horizontal asymptote. I'm going to move that up one, and I'm going to put that dotted line in right here. And as we've indicated before, the asymptote is not officially part of the graph. It's merely an aid in helping us sketch it. If I don't put that dotted line in there, it's not clear that the graph is supposed to be leveling off. So my new origin is one unit above the original origin. Now from this point, I go up one. And going back to the new origin, if I go over one, I go up four. One, two, three, four, which actually puts me across from five. And from the new origin, if I go to the left one, I go up one fourth. Because you see, basically, I'm plotting 4 to the x. I'm just doing it one unit higher than I did before. And my graph comes down like this. And it approaches this horizontal asymptote. And this is the graph of f. Now I hope this sounds very familiar. And you're saying, oh, this is just more of the same old thing. Because it really is. You see, college algebra focuses primarily on drawing graphs of functions. So it's a very visual sort of course. Whereas intermediate algebra dealt primarily with algorithms like the quadratic formula, factoring, polynomials. So it was a different slant on algebra. And what we're doing here is actually analytic geometry when we're drawing graphs of these functions. OK, let's go back to that screen and graph the second function that was listed there. We have g of x equals 0.2 to the x plus 2 power. OK, let's look at the function g of x equals 0.2 to the x plus 2 power. Another way of expressing this is to say this is 1 fifth to the x plus 2 power. And another way of expressing this is to say this is exp, that is the exponential function, base 1 fifth of x plus 2. And the exponent goes inside here. So all of these are referring to the same function just by different names. Now when I go to graph this, I'm going to be making a shift in the function 1 fifth to the x. Which way will I be shifting this function to the left? To the left two units. So I'll move to the left two units at negative 2. This is my new origin, but this is not a point actually on the graph, just where I'll begin to locate the points that are on the graph. Let me just fill in the rest of this axis here. x-axis, y-axes. So at negative 2, this is the new origin. I go up one point. If I go to the right one, I go up one fifth, because the base is one fifth. And if I go to the left one, I go up the reciprocal of one fifth, so I go up five. And it looks like five is right there. So when I graph this function, it comes down, levels off, and it's indistinguishable from the x-axis at that point. Let me ask you some questions about this graph. Are there any x-intercepts? No. There are no x-intercepts, because as we just said, this graph approaches, but it doesn't cross the x-axis. Looking at it from another point of view, if you're looking for x-intercepts, what you do is let y be 0. And if I put a 0 in there, 0 equals 0.2 to the x plus 2 power, this has no solution. Because if you take 0.2 to any power, you will get a positive number. You'll never get 0 for the answer. So there's no x-intercepts. Now, what about y-intercepts? Is that x equal to 0? You said x equal to 0. So this time, I'm going to substitute it in 0 for x, and I get 0.2 to the 0 plus 2 power. And that's 0.2 squared, which is how much? 1, 125th? Well, let's see, 0.2. Well, 2 times 2 is 4. 1 to the 25th. I'm a 1 over 25. OK, 1 over 25 is a fraction. Sure. As a decimal, I was going to say it's 0.04. And Stephen's giving us the answer as a fraction, 1 over 25. Yeah, that's the same thing. And so that tells me that we cross the y-axis right here at 4-hundredths, almost 0, but not quite. OK, let's go to our third example on that graphic. This is the function y equals negative 2 to the 2x plus 6. Now, this one has a few tripping points in it that we need to investigate. Let's consider C, which is y equals negative 2 to the 2x minus 6 power. Now, let me ask you, does this mean negative 2 to the quantity 2x minus 6? Or does it mean 2 to the 2x minus 6 with a negative outside? Which one of those is this? Second one. It is the second one. And it's merely by agreement in notation as to how this is interpreted. We don't put the negative 2, the negative on the 2. We raise the 2 to the exponent. And when we're finished, we put the negative outside. So this is how we interpret that. But it's only by agreement in mathematics that that decision is made. Now, when I graph it, the first thing I'll do is factor the 2 out of the exponent. That's 2 to the 2 times x minus 3. And I remember that that negative is outside the parentheses. It's like I have parentheses right here. But if I show them, it makes it kind of awkward to look at. Now, the next thing I'll do is I'll take the 2 and separate it from the x plus 3 and write it this way. 2 to the 2 squared to the x minus 3. Now, you might say, wait a minute, is that legal? Well, one of our properties of exponents that we looked at on our very first graphic today was that if you raise a to the m power to the n power, that's a to the mn power. This is the product of the two exponents. So what I've done is to take the product and separate it and put a 2 in and the x minus 3 out. Now, what's the advantage of writing it this way, Matthew? Well, you can just raise 2 to the second. OK, which is 4. So why don't we just call it negative 4 to the x minus 3? So this function has undergone some changes, but I think it's easier to graph in this form. So if I want to graph that, what I'll do is make a translation. Which way should I translate this? Up or down or left or right? To the right. To the right, yeah. This says, move it to the right. I'm going to move it to the right 3. And the negative sign tells me the graph is inverted. So rather than going up 1, I'm going to go down 1. And if I go to the right 1, I should go down how much? 4. 4, yeah. I should go down to negative 4. You see, normally, if I went over 1, I would go up the base. But we've inverted it, so I go down the base. So I go down to negative 4. And going back to the new origin, if I go to the left 1, I should go down the reciprocal of 4 or 1 fourth. So my graph looks like this. Yeah. So we have just graphed the function y equals negative 2 to the 2x minus 6. Although we didn't graph it in that form, we graphed it in the form that we wrote up here. OK. So if you have a coefficient in the exponent, like a 2, in this case, I would say bring that 2 in to the base, change the base, and make the exponent look simpler. Let me just work another example like that. Suppose we have f of x equals 1 half to the 3 minus 3x power. Let's see, there are other ways we could express this. We could say this is the function exp sub 1 half of what? What would I put in the parentheses? 3 minus 3x. Yeah, I could write it that way, although it certainly doesn't look any better than it did to begin with. I think part of the problem is I have a coefficient on the x. So let's try going this route. I'm going to factor out the negative 3 and call this x minus 1. I factor out a 3 because 3 is a common factor, and I factor out a negative because I'd like the x to come first rather than second. Now the negative 3, I can bring inside the 1 half, and that's 1 half to the negative 3 power, and then that's raised to the x minus 1 power. Now what's 1 half to the negative 3? 8. That would be 8, yeah. That's 8 to the x minus 1. Okay, now I think we actually have something we can deal with fairly easily. This is the same thing as what I had over here. So what I've done is I've used properties of exponents to get me from there to 8 to the x minus 1, and now 8 to the x minus 1 I figure I can actually handle. That's not such a complicated graph. Okay, let's graph it right here. I'll set up my axes, and let's see, was there a question? No, okay. So if I go over 1, that's my new origin, and I should go up 1 to locate a point. So at 1, I go up 1. If I go over another one, I should go up how much? 8. 8, I should go up 8. Well, for the sake of argument, let's say 8 is right there. And if I go back 1, which puts me right at the origin, I should go up how much? 1 eighth. 1 eighth, right here. And what we have just graphed is the function f of x equals 1 half to the 3 minus 3x power. It looks very ugly that way, but when we say it's the same thing as 8 to the x minus 1, it doesn't look so difficult in that format. So we've graphed both these functions at once, because they're the same thing. We've graphed 1 half to the 3 minus 3x, and we've also graphed 8 to the x minus 1 power. It's the same function going by different names. Okay, now I'd like for us to look at one special exponential function that has numerous applications in business and chemistry and physics engineering, and in still other disciplines. And this one is called the natural exponential function. Now, the base of this function is a special number called e. If we can come back to the green screen for a moment, let me just show the class something here. You know, there is a number 1.41428, something like this. And nobody remembers those decimals exactly. In fact, I'm not quite sure I have it exact, but the way we write that number is we call it the square root of 2, because this is a symbol for an infinite decimal expansion that we just couldn't write out. It's infinitely long. On the other hand, 1.732 dot dot dot. Does anybody know what that is? Square root of 3. That is the square root of 3, or at least it's the first few decimal expressions, decimal values of the square root of 3. How about this number? 3.141592, what number is that? Pi. Pi, yeah. See, we have a symbol for that number, because it's just too difficult to try to express it exactly. In fact, it's impossible to express it exactly, because it's infinitely long. We don't have enough time in our lifetimes to write out a number infinitely long. Well, there is another number that you haven't seen before, but you'll see now is 2.718 dot dot dot dot dot. This number is called E. Now, the reason you haven't seen it before is because this number is only useful as an exponential number with exponential functions, and now we have arrived at that point. So this is sometimes referred to as the natural exponential number, or I'll just say E for exponential number. Okay, so what you want to remember is the 2.718. Okay, now let's go back to this graphic, and we will look at what's called the natural exponential function. It says the natural exponential function is the function f of x equals E to the x, where E is approximately 2.718. And my example here has two parts, and it says sketch the graphs of the following functions. First of all, let's graph the natural exponential function E to the x, and then let's graph the function E to the x minus 1 minus 2. Now, these are as easy to graph as the last ones that we just looked at. Let's take this first case, f of x equals E to the x. Now, you might ask the question, why is it called the natural exponential function? What's natural about 2.718 dot dot dot? There's nothing, doesn't seem natural at all. Well, this function arises in a, quote, natural way when you investigate certain applications, applications that we'll be seeing in the next few episodes. So because of it, it's sort of a natural representation of certain problems, it's called the natural exponential function. And you might say, Dennis, how natural is it? Well, I tell you what, it's so natural, if you look on the green screen, this number is, this function is sometimes abbreviated as exp of x, and there's no subscript. If there's no subscript on the exp function, they mean to use the natural base, and the natural base is base e, not base 2, not base 5, base e. So if you wanted, if you wanted for emphasis, you could put a little e down there, but that would be redundant because if you leave it out, it's assumed to be the natural exponential function. Again, I will not use this notation, this exp notation on exams or homework, but I'm just letting you know this for applications of this material. Okay, I'd like to graph this function, and we said that e is approximately 2.718. So when I graph this exponential function, I'll graph it just like I've graphed all the others. There are three target points. What would be the first target point? Zero one. Zero one, exactly, nothing's changed. What would be the second target point? One e. One and go up e, okay? Well, that's about 2.7, which kind of approximate that. That's about e, let's say, right about there. And if I go back one, I go up one over e. Now, you can try this on your calculator. If you take one over e, that's approximately one over 2.718, and that's approximately 0.36. So let's say it's roughly a third. I'll go up roughly a third because who's going to be able to tell the difference between 0.36 and 0.33 indefinitely? So I locate these three points and I draw the natural exponential function. Well, it should be approaching the x-axis right there. Okay, now, this is the first time we've graphed this function, but it won't be the last. You will see variations of this graph come up frequently in the rest of the material for exam three, but you'll see it come up in other courses where college algebra is applied. So when you look at this function, it looks very much like all the other exponential functions that we've been graphing, except it's allowed us to discuss this new number, e. Let's see, there was another function there that we wanted to graph. G of x equals e to the x minus 1 minus 2. Let's graph that one. G to the x equals e to the x minus 1 minus 2. First of all, can anyone think of a way to write that using the exp notation? exp of x minus 1, close parentheses, minus 2. Minus 2. Yeah, it looks kind of weird that way, doesn't it? But believe it or not, that is another representation for this function. Okay, we'd like to graph this. So what I'll do is to make two translations. I'll have to move one to the right and two down. So one to the right and two down. Now, when I go two down, I have to take my horizontal asymptote with me, but I didn't move a vertical asymptote over because there wasn't a vertical asymptote. There's only a horizontal asymptote here. My new origin is sitting right here because my origin moved over one and down two. Now, from the new origin, I go up one to locate the first target point. From the new origin, I go over one and I go up e. Let's see, I better fill in a few values there. I go up e, 1, 2, 2.7. That's actually e minus 2 on the y-axis. That's e minus 2 because I started from 2 below. And if I go to the left one, I go up 1 over e, which is about 0.36. And now, I draw the graph. You see, it's really very fast. And we've now graphed e to the x minus 1 minus 2, or you could say we've graphed the exp function of x minus 1 minus 2. Yeah, that's all written on one line there. That's not an exponent. I should maybe move that up a little bit. Yeah, okay, that's all there is to that one. This one is not on the graphic, but let me just ask you how you would graph this. Suppose you wanted to graph the function f of x equals 2 times e to the 1 minus, let's say 1 plus x, e to the 1 plus x. Excuse me, let me just change that in one way. I'm going to change that to the negative x. I think that would be more instructive. Any suggestions on how we would graph this? We can make the 2 times 1 over e to the x. Right, this is 2 times 1 over e to the x. What Stephen has done is he's moved the negative into the base and he inverted the base. That's fine. So now I have an exponential function with a base smaller than 1. So there's no translation, vertical or horizontal. I start off and I go up. Well, normally I'd go up 1, but there's a stretch of 2, so I should go up 2. I should go up to 2, because we've stretched it. And if I go to the right one, I should go up 1 over e. How much do we say 1 over e is? About a third. It's about a third, so I'll go up about two thirds, roughly. And if I go to the left one, I should go up e, 2.7. I'll go up 5.4. I'll double e. 1, 2, 3, 4. Let's see. I'm going to have to move that out of the way, Stephen, to get your graph in here. 5, 5.4. I'm going to have to move the whole thing out of the way. Sorry about that. 6, 5, 6, 5.4 is right about there. And my graph looks like this. This is the graph of f of x equals 2 e to the negative x power. Okay. In the last few minutes here, I'd like to compare two graphs on a graphing calculator. Okay. For our last graphic, let's look at the battle of the titans, you might say. We want to look at a comparison of the graphs of exponential and polynomial functions. So on this graphic, it says a graphing calculator helps us to demonstrate that polynomial functions claim fast, but nothing like exponential functions. And we want to compare the graphs of f of x equals 2 to the x and x cubed. Okay. So if we come to the green screen here, I've gotten this set up so that we're graphing, first of all, 2 to the x. That's the function of y1. And then x cubed is the function of y2. Now, for example, what if I substitute in x equals 1? How much would be 2 to the x if x is 1? 2. It's 2. And how much would be x cubed when x is 1? 1. So it looks like the exponential function is bigger than the cubic function. But if you go to 3, if you plug in 3, 2 to the third power is 8, but 3 cubed is 27. Now the cubic function is bigger than the exponential function. Let's take a look at this graph. I've set up the window. Let me just show you the window size I've chosen here. My x's go from 0 to 5, and my scale is one unit on the x-axis. And my y units go from 0 to 10. And just so it's not too cluttered, I made the scaling every five units. And when I draw the graph, here's the exponential function, and then here's the cubic function starting underneath, but it crosses it. So it looks like the cubic function passes up the exponential function. In fact, it certainly does. But I think the exponential function is going to crisscross with it, and the exponential function will be on top eventually. Let's zoom out. I'm going to pick a larger window. Okay, this time let's have our x's go from 0 to 10. And I'll change my x scale so that, let's say, every five units, there's a tick mark on the x-axis. And I'll have my y's go from 0 to 200. In which case, may we better have our y scale go every 50 units. And let's draw this graph. Okay, so first we see the exponential function 2 to the x turning up. And now we see the cubic function, and you see here's the cubic function on top. It was a graph second, because it was the second one in my list. And they did a little crisscross back here that we just observed, but now it's so compressed we can't see it anymore. The cubic function is above the exponential. But I think there's going to be another crisscross higher up. Let's go back to the window and raise it. I'm going to have my x's and my y's go up to 1,000. And let's have our y scale be every 250 units, and let's see what happens. Here's the graph. Here's my exponential function. It's rising much slower now because this is 1,000 unit high. Look, they're crossing. I think if I go up a little bit higher and go over a little bit wider, we'll actually see the crisscross. So I go to the window one more time, and let's have this go from 0 to, let's say, 15. And let's have this go from 0 to maybe, let's say, 1,250. And the graph looks like this. Here's the exponential function. Here comes the cubic function. And they cross, so the exponential functions back on top, they will never cross again. There were two intersections. There was one very close to the origin. There's another one further up here. And that's all for today. We'll have to stop.