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.  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.  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.  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.


I'm going to do a demo, and hopefully it'll be just fine. So what I'm hoping to show here is an app, a front end and a back end app. You might have seen this Fortune Teller demo before. It's a spring app, but we converted it to Ruby because I like Ruby. And what I'm hoping to show today is connecting your front end app to your back end app securely using some of the new features on the platform and then updating your dependencies in a zero downtime manner. So let's see if that all works. What you can see over here, I'm logged in here. We just have our Fortune Teller UI and our Fortune Teller server. You've got two instances of the server, three instances of the UI. And the UI has a route map to it. The server doesn't yet. And you can see our UI is over here. I'm going to click this Auto Refresh button. And you can start to see that hopefully it should cycle through the different instances. And it is not cycling through. All right, that's unfortunate. Let me try pushing that again and seeing what's going on there. So CF push, my Fortune Teller UI with the Ruby build pack over here. OK. Well, that's working now. We are now cycling through the different instances. We are not cycling through the different instances, I'm sorry.  Smash left. This is over here. I pushed it. And it's waiting for the app to start over here. If this doesn't work out, I'm going to go to the video. But I'm going to try for one more minute because it's a really cool demo. All right, that's up. It really doesn't seem to want to be refreshing. Well, let's see how far we can get with it. And then I can skip to the video if needed. So you can see here, if I want to get the front end app connected to the back end app, I want to create an internal route for the back end app. We have with recently introduced Polyglot Service Discovery, there's this new internal only domain. It's not externally accessible, apps.internal here. And I can map that, create a route, and map it to my back end app using that internal only domain. And that's it. It's mapped. You can see it with CF apps. You can see that Fortune Teller server over here. And then, but we still need to teach our front end app where that back end app is. We can just, there's many ways of doing that, but we can use user-provided services, tell it where that back end app is on, apps.internal. And then we can bind the service here. And at this point, you usually have to restage or restart your app to pick up that new configuration. But actually, you can do a zero downtime restart. And because the app isn't refreshing very much on the side where it looks like it is, why is it doing this? Then we can actually do the zero downtime restart. And I wish the app would work properly for that. But it actually goes through and restarts instance without taking your app down. You won't have to do a blue-green deploy to achieve this configuration change, which is really cool. Moving on, the front end app still can't talk to the back end app, because we haven't actually given it the network policy, secure network policy, to allow the front end app to talk to the back end app. So we can just add that. And this is a feature of container networking. So we can add that network policy. You can see that policy defined over here, connecting the Fortune Teller server and the Fortune Teller UI. And if this was cooperating with me, which I don't know why, it's not there. It worked, trust me. It really did connect only on that network policy part. And you've got more fortunes. That's so awesome. It's only just now working. All right, on to the next Friday I wanted to demo, which was updating your dependencies. And I hope you all know that Ubuntu, they will be end, they're moving on. They support their things for five years. The trusty version, the thing that CFLinux FS2, the stack is based on, will no longer get security updates as of April 2019. So you have a limited amount of time to migrate your apps using that stack from CFLinux FS2 to CFLinux FS3. And this is what I'm about to show you is very experimental. The CLI with a hidden command just became available yesterday. But the build packs are available, and you can start trying them out. So here you can see, actually, we have a Ruby build pack that's a CFLinux FS2 stack and another one that's CFLinux FS3. And our apps over here, they're actually using 14.04, that's CFLinux FS2. And we can actually do a zero downtime update and update the stack at the same time. So let's do that. And hopefully, this will also, this too, shall work. So other things to note here, you don't. You'll probably want to make a plan, enable your clients to slowly opt in, then change the default, and then force them all over. And you can hopefully, by the time you need to force them all over, you can use this and do it very safely. And it's a great way to just update your dependencies, stay secure, stay up to date. And as you'll notice with the zero downtime update, your app is not going down. I didn't have to do a blue-green deploy. The staging, usually, when you push again, your app will go down. It'll stage. That might take anywhere from five to 10 minutes. And here, the app is just working fine, cycling through. At some point, you'll see as each one starts, the stack will be updated to Ubuntu 1804 by Bionic Beaver, which will be good for five years. And you'll hopefully start to see that soonish now that it's waiting for the app to start. And it'll go through one by one. Did it happen? All right. Well, it's going through. There's one. There's another one. And that's my demo. Thank you very much.


widespread voter apathy and vote-buying marks the local government elections held in Lagos and Oregon State on Saturday. And you've done nothing deserving of commendation. PDP Publicity Secretary Kala Lovadeon tells Governor Yahya Bailo of Kogi State. Well, this is Plus Politics and I am Mary Anifor. White-spread voter apathy and vote-buying characterized the local government elections held in Lagos and Oregon State on Saturday. Reports from different parts of the state show that many voters shone the polls for lack of interest in the exercise, fear of electoral violence and lack of faith in the outcome of the election. For instance, the presiding officer at Ujrodu Grammar School 1, polling unit 17, Ebel Obinna, said only seven persons had cast their votes as of 11.15 a.m. out of the 150 registered voters. In most treats in the councils that were visited, use of voting age was seen playing football while major roads were emptied due to the enforcement of no movement. Well, joining us to discuss this is Plus TV Africa's correspondent Aneta Felix and Jacinta Obuco who visited the polling areas and of course later on we'll be talking with Chike Chude who's a political affairs analyst. Thank you very much ladies and gentlemen for joining us. Thank you. Good evening, Mary An. It's a pleasure. Great, great, great.  You were out on the streets on Saturday. I mean, I saw a news this morning of a PDP party chairman in the state at the time he showed up to cast his votes. He was the only person that had casted a vote at that polling unit showing that many people were not even interested in the election. But give me a feel of what you actually experienced and why you think people did not show up for that election. Okay, I think we should begin with the latest deployment of the material. The election was supposed to begin at 8 a.m. and end around 3 p.m. on Saturday. But what happened was that when we visited the Ateosa local government for Balindi on Saturday, we found that as at 9 30 a.m., full and addict staff of the Lagos State Independence Electoral Commission were still receiving election materials and they were actually jam-packed outside the gates of the secretariat waiting for buses that were designated to take them to different polling units in Ateosa local government area. So I think that really is the beginning of the issue that we saw regarding the logo to turn out. So it's possible that maybe people had gone to their different polling units at 8 a.m. but were still receiving their materials. So they definitely were nowhere to be found. So when eventually these buses came around and these electoral buses, they consumed your screen with the plastic ballad boxes with the card readers, when they eventually set out to go there, they had to begin to set up and ensure that all security measures were in place. And then what next happened was that I visited award in EJ. It was award 10 polling unit two in EJ in Ateosa local government area. When I got there that morning, I found that there were less than 20 people who had come out to vote. I had assumed that it's early in the morning people will still come out much later. But what I found shockingly was that rather than the crowds to increase, it instead declined. So I took a stroll into the estate and off in the state and I found that that it seemed like a normal Saturday. People were around on the streets, people were having fun, talking with neighbors. It didn't seem like there was an election taking place a few steps away from them in EJ. There was a primary school that was being used for the election. And it really makes me ask questions. Where did all our sensitization go? I mean, I co-hosted the breakfast and pasta in the Africa and we've had stakeholders in politics who have representatives of the independent national electoral commission, INE, come out to educate voters, educate Nigerians because people didn't really seem interested. And then when I discussed with security experts, political experts as well, the vibe I got from them was that a dissatisfaction by the public as to what's happening in the country probably is responsible. You see that people feel that they're not being respected, they're not being valued by the government. You see the security issues in the country. So it seems that people simply were just, you know, displeased with the government. And maybe also we can add to, maybe we didn't get enough sensitization of people, maybe people didn't really register to vote, even though according to the statistics we have, there are about 6 million registered voters in Lagos, 13,323 polling units. But across Etiusa, across Lagos, the polling units were scanty. I mean, begs the question if, like you said, orientation was properly done because I remember there were people who were still asking us at Friday if the elections were going to be on that weekend or it was going to be this week that we're in. So meaning that the Lagos state CIEC probably didn't do its job or maybe again it could be that people really don't care about the local government election. But I'm going to come back to UNS and let me go to Achike. Achike, I'm curious to understand why we complain so much about government not doing this or not doing that but when it's time for us to make decisions by voting, coming out to put a stamp on what we want government to do, we're nowhere to be found. And I keep pointing to the fact that we're very good at tweeting and posting on social media but most of us don't have valid voters card. Something as simple as local government elections, we didn't see people show up for it and is this enough to just say that we are not happy with the government? Is that enough a reason for us to not show up to the polls? No, I do not think it's enough reasons though that it's a reason but definitely cannot be enough reason especially when you know, I mean, ultimately that our ability to get involved in the political process goes a long way in putting some measure of pressure on the political actors to get them to do things that under normal circumstances they will not have done but simply because the people that they are watching and the people are also involved and the people can play a role in them getting elected or re-elected. So and somebody has made that argument before that politicians usually like this level of apathy because it is, they find it much easier to manipulate the process when the people are not greatly involved in their in their own politics. But how many Nigerians know that? Because you see you just you just went to the next thing that I was going to point out too. There's a video that surfaced on social media it's been making the rounds of a woman who was you know, Tom printing on different ballot papers. There's so many questions to that video. Where did she get those ballot papers? There obviously means that somebody within the Lagos State SIEC had facilitated those ballot papers getting into her hands. It means that the SIEC in the state is also complicit. Results in that area wherever it was needs to be recounted or something needs to be done about it and how many heads do we make sure get rolled. So yes there are people who take advantage of the fact that people don't show up. So let's Tom print and then you see a large turnout. Meanwhile in other places there was nobody. So but how many Nigerians realize that they're losing and these politicians in quote are winning? Yeah I think you have put it succinctly. I want to believe that there are so many Nigerians again and that is that the world we find ourselves in a season season of anomaly where those things that should matter to the people do not exactly matter to them. Where people are not able to make the connection between the life they are living now the lives they will live in future and that the politics of the day that determines the quality of life that they are going to live so they have not been able to make that connection and that I think that that is the is your fortunate situation and it's not even as safe that they do not realize that when they get involved you know with the right kind of you know politics that things can change. But if you look at I mean the order of elections in this country the fact that you have elections at the national level, presidential level, governorship level you have elections to the various you know houses of assembly as well as national assembly. You realize that in the scale on the scale of importance that you could again at the local government cancel elections which under normal circumstances because of its proximity to the people and because of its ability to touch impact more directly the lives of the people. But you will find that paradoxically it is the least important of the elections and so when you weigh most of these elections that are going on in this country in terms of notoriety you know you find that the most notorious in terms of the outcome of a local elections in the country is the local government election. Why do I say so? It is very clear that things are not working exactly well across the length and breadth of this country at the local government levels. It's not just legal state, particularly in all the states of the country. So when you talk about the you know failure of governance you also have to start from the grassroots and that is the local government elections. But when you look at it you consider the fact that you know most of the local government and that is the consensus among most Nigerians across the states, across political party lines that local government you know the chairman and the councilors are not doing much to change the lives of the people. Then the question you ask yourself is how come that in virtually every state in the country where local government elections are held there is a 90 to 100 percent you know success rate for the party for the political party in charge of the states in those states. That's a whole that's a whole that's another kettle of fish on its own and I want to revisit it you know we will revisit it but we're being joined by one of our correspondents um Jacinta Obu who also was in town to cover some of the proceedings of the elections on Saturday and Jacinta can you hear me? Yes I can. Great just give us an idea of what you experienced while covering the elections the local government elections on Saturday. Okay so I didn't get that but I assume you are asking what I observed during the election yes okay yeah so the parts of the areas we observe in meagres some part of meagres uh I could tell it's speechful I studied from a joey state although um the first one the first polling unit I got to um the deciding officer there complained of the card reader of them not working and we saw him putting a call across to get it verified and um people were already we got to explore people are already queuing waiting for the whole process of expression with the whole um delay and uh so we left that to first start first start through or speechful but just no turnout there then from there we went to ketchup and uh we we met with um one and this pitty pitty chairman that was um um and in the state pitty pitty chairman there and he complained bitterly um over the turnout of voters and that could tell and he said that could tell how people don't believe in the electoral process anymore in the country and um he as well called on his counterpart political counterpart to better know that um this is going to work for them and about them for about the people so as well we moved down to us born uh polling unit there where we witnessed um people and we saw some agents political editors there so in a complaining of missing polling units there as they said this could suggest um election region and all of that so but generally I can tell like what I witnessed it was peaceful except for the low turnout of voters in terms of the the the missing polling units did you probe forward to ask the siac members if you know iNEC had made them aware of these polling units not being available or it was just an oversight uh by the siac did you find out yeah yeah we we tried finding out from the the officers we met that they just said they don't know they were just sent there to um to their own part they don't know about the other uh polling unit I'm missing but the agents there said they are supposed to be two in that area like two in different areas there but just one one and they can't see the a senior uh political and a senior iNEC official to ask uh where about the polling unit so the other ones that if they want the very ones that are polling with their officials said they don't have an idea why um others are missing and wow it was a constraint for uh voters there and as well the agents there well Dostinto Bikku is one of our correspondents here in Plos TV Africa thank you very much Dostinto for speaking with us I'm going to go back to um I'm going to go back to Annetta because Annetta I remember we also uh Plos TV had a trending video of a woman who was given a thousand Naira um to vote uh what particular polling unit was that and what area that was at ward 10 that simple unit two that I mentioned that I went to in EJ that was you know just before the entrance of the estate um Dolphin and that's at equal your body in the ATSA local government area so I had been hearing that you know party agents were distributing a thousand Naira each you know if you voted for their party well I said I needed to get evidence of that and I watched closely and I you know observed that indeed some party agents whom I had personally met and spoken to you know at the secretariat you know were actually giving up how many I brought on my phone immediately started to capture that and I saw you know party agents giving a thousand Naira to women then I went to one of the women who received that one thousand Naira as you're seeing on your screen I went into her and I said oh I see your collecting a thousand Naira I'm simply for voting that why did you do that why did you decide to steal your vote and her answer was I don't know I really thought about the answers her answer since Saturday asking myself is this really what poverty is you know has made of us as a country and I remember that some other party agents had gone there to complain to the security agents the police officers who were there and the response was that whatever happened outside the gates of the pulley unit was not their concern that they were deployed to the pulley unit to make sure that the operations of the election was smooth and whatever is happening outside the gate was under the business and indeed this was happening just out near the gates now if you take a look at that video that I sent you and you find that these party agents first of all the school has a low fence so these party agents I don't know if maybe they find a way to talk to the people they bait them and then they overload the low fence and then once you come out the hundred thousand Naira it's just so sad I mean the reaction we've gotten on social media has been massive people even saying oh they've even doubled the money like some elections ago they were distributing 500 Naira and we know this is nothing new in Nigeria it's just the hard evidence that seems to be lacking when we talk about the stories you know the fact that I was able to in the spot of that moment bring up my phone to capture when that woman was receiving a thousand Naira and interrogate her you know so we know that this is something that happened Nigerian party agents you know distributing money simply to buy votes so you say you ask the question that is this where poverty has led us as a people but do you really think it's poverty or is ignorance because if I mean yes poverty can be you know it can be a problem but really could it also be ignorance at play because she probably doesn't even know and that's her answer and she doesn't even know the consequences maybe I'm saying maybe she doesn't even know the consequences of that one thousand Naira that she's taken and whatever party that she voted for exactly Marianne one of the first questions I asked you know when I was bottom was where did all our sensitization go so it then seems like we're not seeing a lot maybe we are over 200 million and it seems that our voices you know are not loud enough our voices are not reaching the ears of people they supposed to reach because I mean look at an election as basic as a local government election I mean how bad can he get it yeah we're talking about a local government election people who are coming out for champion championship positions championship candidates councillor and you know aspirants and all of that these are people that should be closed close to the grassroots these are people that you know the community should know these are people that should be known in the community these are people that should be influential these are people that should have already been doing things because I mean if you know of your community audience I think we're having connection problems with Anessa so we'll just let her go for a second if her connection is great we'll bring her back but back to you at 2k today Anessa has raised some of the issues I wanted to raise how many of us really understand that the government that we need most is the government that's closest to what's the most and I'm talking about local elections I'm talking about local government because we as Nigerians seem to somewhat accord the responsibilities of our local government to states and mostly the federal so the gutter in your area is blocked or or maybe you have a problem you know a local government problem and then you say Buhari is not doing his job Bwari is bad Bwari must go that is also a you know a result of the ignorance on our part but when these people who are running for offices whether they belong to party A, B, C, D or E how many of these people do we know how many of these people can we say we understand their track record or can vote for to even vote for because I mean really it's one thing to come out and vote it's another to know who you're voting for especially in Lagos state do we really pay attention to our local leaders do we want to know them or we're more interested in Governor Sao Anu? Well I think it's it speaks for the problems that we have in this country and it's all over it would appear that there is a sense in which it would appear it will seem that the people have given up you know when it comes to governance and people are no longer interested in what happens you know with regards to the politics of their country so they all believe that the only way to get by is just to face those things that matter most to them but they do not see the connection between like I've said the policies that are you know taking that have been dished out on a daily basis by government officials and the way their lives turn off and so this has been a major issue it's the inability to understand that connection that they that they ought to be much more involved than they are that they have been involved you know in the politics of their their country especially the you know their the politics within their immediate environment I think that that has been the you know issue and and again hello can you hear me I can hear you hello can you hear me yes I can okay so that has that has been the problem but but beyond that again is a fact that yes we know that to some extent poverty has played a very terrible role in manipulating the Nigerian people politically but at the same time you talked about ignorance ignorance has also played its own made its own share of a play its own you know role and as well as you know sentiments on on due sentiments too especially when it comes to national elections but beyond that it would also appear and that is one thing that a lot of us are are failing to grasp and that is that that the political parties are the ones that have a duty to ensure that they are that voters come to vote for them because they are the greatest losers of an election that goes badly for them and they are the greatest you know beneficiaries of elections in which they have you know play the role in which they have won so everything about election campaigning it's about how to bring the people to the polls to the polling stations and so if they ultimately at the end of the day it comes to election time and you do not have the people coming to the polls to vote it simply means that both politicians and the politicians and the political parties have failed in the ability to get their messages across to be able to get the people to the polling stations to vote but perhaps there's a reason for that perhaps they believe that they don't need to spend so much money because ultimately they know that the results of the elections have been written and and and that is ultimately the you know the mindset of of the people and that's why I did ask the question you know I made the observation that in virtually all of the elections that take place especially at the level of local government councils about 95 percent or 100 percent of of victories in those elections you know go to the party in power you know the party of the day but who's responsible who's whose responsibility is it to make that change because like you are one of the electorates obviously you are one of those people who's complaining that oh if the PDP is in power the PDP sweeps the local government I mean literally happens across all the states of the federation for the state governors who've decided yeah but but but but the owners who where is the owners who who has the responsibility of changing that because if if like you just said if if political parties take it upon themselves to not just campaign when it's time for campaign season especially at the local levels acquaint yourselves with the people in those areas so they know who you are and not what you're about but who you are and and then they now decide what you're about and decide if they want to vote for you but we only see these people when it's time for elections and then you hope that people would vote for you because maybe the government of the day is not doing right you see obvious we see the issue you expect magic to happen you know you're absolutely right so but what it means yes I agree that that should be the thing you know because it is because it is and the government that is closest to the people so the possibility there's a greater possibility that the people would be in a better position to know them they see them on the streets they know their children and all of that but obviously that is not happening and the politicians are not even reaching out to them street to street to talk to these people and why are the politicians not doing that because they also know they also enjoy the apathy of the people because ultimately whether out of you know an electorate of about 100,000 maybe 50 people 100 who cast their vote they are going to be declared as winners of that you know political process so it doesn't really matter to them even if they don't spend as much as they used to they used to spend like that's why I say that ultimately voter apathy is more beneficial to the politicians because they make a lot they make a lot of you know they take a lot of mileage you know from from from the apathy of the people but ultimately I agree with you it is the people themselves they are the greatest stakeholders in the political process because they are the biggest victims of an election that has gone wrong you know and then if if if they have competent you know people in power they are the ones that benefit you know also from that process but the fact that they are not doing that also shows that the people and themselves are not responsible and so what you have is an abdication of political responsibility by the people themselves and ultimately if there is a you know a price that will be paid the people themselves are going to pay the price so when we already we're already bringing that price on to it oh yeah we are because I mean people talk about talk about about the cost of foodstuffs in the country people talk about bad roads people talk about insecurity people talk about but you are not doing anything to put pressure on the people who have a constitutional mandate to get these things done you're not doing that you are abdicating your responsibility and then the next thing is also holding responsible where things do not go well but the people perhaps are even much more irresponsible than the politicians yeah well Anessa is back Anessa in closing you and I obviously are representatives of the fourth estate of the realm and half the time when you know when it comes down to it you know the media is pointed at and and they say oh you're not doing enough to sensitize the people we and then we also know that NOA comes alive just quarter to the elections which is not enough time for the kind of voter education that we're looking at but we also noticed Anetta I'm sure you covered the elections in 2019 and the level of you know low turnouts wasn't as bad as this obviously but you know could this also be a you know a pointer to what we might be expecting in 2023 with the you know the dampened spirits of the average Nigerian voter what are your expectations Mary Ann you took the words right out of my mouth I was thinking to myself just before this meeting asking myself what witnessed during you know Saturday's local government elections could this possibly be a pointer to what we're about to witness in 2023 and I fear for our political future because if you know youth do not come out to represent themselves in the electoral process come 2023 I really wonder what our future would look like because we continue to complain on social media about the lack of infrastructure about just the death of everything in the country we got in politics so you know our socioeconomic reality everything education and I like how Mr I think that Anetta it's frozen again unfortunately Achiketure I want to thank you but we have to go we're almost out of time. Achiketure is a political affairs analyst and Anetta Felix is a an anchor of the breakfast show here on Plus TV Africa and she's also a correspondent once I want to say thank you to Justin Taubeco who is also one of our correspondents here in Plus TV Africa thank you gentlemen and ladies for being part of the conversation. It's a pleasure thank you. All right well we'll take a short break and when we come back we'll be discussing the performance of the Kogi state governor and of course what the PDP publicity secretary has to say about it but yes we will be joined by the Honorable Commissioner for Information in Kogi State. Stay with us.


Okay, so in this step what we're now going to do is we're going to create a hierarchy So we're going to make all of these planets belong to the sun so that whenever we move the sun everything else will go with it And it'll also be set up perfectly for animation when we get to that So what I'm going to do is I'm going to go to windows and outliner And what the outliner is is it's a little window that lists everything in your scene So you can see all the geometry we've created so far is listed If you've forgotten to name anything, name it now And there you can see there's my sun, there's the planets, etc And what I need to do is make this planet, so the first planet, belong to the sun So again, there are multiple ways of doing this The first way is by using the middle mouse button, that one And I'm going to click and drag with that the planet onto the sun and then let go And you can see a little plus symbol appears and when I click on that You can see that there is now a relationship there, the planet belongs to the sun So whenever I now click on the sun, it also selects the planet, which is very useful Okay, I'm now going to parent planet one to the sun So I'm going to click on it, so I'm just showing you a different way of doing it now And now I'm going to control click, no, shift click, sorry, the sun And now I'm going to press P on the keyboard and P is the keyboard shortcut for parent And you can see that now planet and planet one both belong to the sun And now what I want you to do is parent all the planets to the sun And then parent the ring to the planet that belongs to and the moon to the planet that belongs to So that your hierarchy is set up so that it looks like a man will in a second Okay, so when you finished, this here is what your hierarchy should look like And again, like I said, whenever you now select your sun, it should select everything else Which makes moving this solar system as a whole so much easier And the reason that happens is through something called inheritance And what that means is that the child objects will always do what the parent object does in relation to the parent object So you don't need to follow this, but I'll show you what I mean So if I was to scale the sun, you'll see that all the planets scale too But they don't just scale on the spot, they actually scale in relation to the sun So the gap, the distance between the planets is shrinking as well So it's almost like they're one object, which is really useful The same happens if you rotate, you can see that they all rotate But they're rotating around the parent object, so they're rotating around with the sun And the same happens with moving as well, which is going to be useful for when we move those planets next