 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. In this lecture, we're going to explore the relationship between the algebraic version of some function, let's call it F for the sake of conversation here. We want to compare the relationship of this algebraic version of the function with its graph, that is the geometric representation of that same function. In particular, we want to see how modifications of the algebraic formula affect the graph and vice versa. What you can see on the screen right now is a table, which is going to enumerate the different types of graph transformations that we want to talk about in this section. Now, it's difficult to show all of this on the screen, all of once in such a way that you can actually read it. My recommendation is that if you want to see this table to actually click the lecture notes, which you can find as a link to this video. That way, on the first page of the lecture notes, you can see this table in complete detail. But what I want to do in this video is talk about the different types of graph transformations we're going to be interested in in this section. These are going to be transformations we revisit over and over again for this entire series. The first type of transformation that we want to talk about is that of a shift. Sometimes they get different names, shifts. Sometimes they're called translations or we're moving the graph. That's the main idea here. And so when it comes to a graph, we can move it up, we can move it down, we can move it left, we can move it right. And how is this accomplished? Well, all of these are going to start off with the standard graph y equals f of x. So when you see this right here, y equals f of x, what I want you to think of is this is a graph that we know how to draw. And this this will often be one of the classic functions that we've seen before, like y equals x squared, y equals x cubed, y equals the square root of x. There's a standard set of function families that we play around with all the time. And so we're going to be trying to do transformations to that. Now, we don't necessarily need to know what those graphs are right now. At this early stage, you'll be given the graph y equals f of x. So you'll be given the picture. We just want to analyze how does that picture change as we add stuff to the formulas to track things whatsoever. So if you have the function y equals f of x, and then you add some number to the function f of x plus k, this will have the effect that you're going to raise the graph by k units. So it's going to go up by k units. So I like to think of this way, often abbreviate this as I draw a little up arrow and draw k. This will show that the graph goes up by k units. And we'll show you a picture of this in just a moment in the opposite direction. If you take y equals f of x minus k, where in both situations, we're considering k to be a positive number because if you subtract a negative, you're actually adding a positive. So we want to think about is are we adding a positive or subtracting a positive? Well, if you take y equals f of x minus k, this has the effect that you're going to lower the graph by k units, in which case I often denote this by drawing a little arrow downward with a k. And so we get this by, you know, we do a vertical shift upward by just adding k to the expression f of x and we shift things down by subtracting k. This is if you want to do a vertical shift. On the other hand, if you want to do a horizontal shift, that is, you want to move things left and right. This is accomplished in a slightly different manner. You take y equals f of x plus h. If you want to shift things, if you want to shift things horizontally. So if you take y equals f of x plus h, what this will do in this case, you're actually replacing the x inside of the function with x plus h. This is an algebraic evaluation. We did some examples of this earlier. We replace each x in the formula with x plus h. And now you have to be very careful here. This has the effect that this will actually shift the graph to the left by h units. So replacing x with x plus h actually does a shift left by h units. h here again is just some positive number. Replacing x with x plus h shifts it to the left. And if you replace x with an x minus h right here. So every occurrence of x in the formula you are replaced with x minus h. This actually has an effect of shifting everything to the right by h units. So again, look out for this. Things with the horizontal shift, students often feel like it's working backwards to the vertical shifts because it's like, OK, adding something to the y coordinate moves it up, subtracting something to the y coordinate moves it down. That seems to make sense. But in this situation, adding something to the x slows it down, you might think, because it moves it to the left and subtracting subtracting from the x square moves it right. How's that is? Well, I would recommend you think about this. You want to think of these as instead, not we want to think of this as some type of like head start or a handicap. We're placing on a racer for a moment. So like if you were racing your little brother, right, he's super slow. You're super fast, right? You got such big legs compared to that little guy. In which case, in order to try to make it a fair race, you're going to give him a head start. And so in order to get a head start, you want to think of it as he's going to start before you. So you are getting sort of like this, this handicap on your race. And so you actually, if the same graph starting to the right, it's kind of like it started after the fact. If we think of the x axis as like an axis of time. So by by subtracting, you're actually kind of slowing it down that it started later. Or if you're adding that to it, it's like, oh, you started earlier. And so it might seem like it works a little bit backwards. And this is something you want to kind of get used to when you work with these graph transformations. I often like to refer to this phenomenon as the horizontal zone. And I also like to think of it kind of like the Twilight Zone a little bit. Do, do, do, do, do, do, do that things inside the horizontal zone. The horizontal zone is everything between the parentheses. You have F of something. Anything in that something we call the horizontal zone. And one thing you're going to see very quickly is that the horizontal zone works differently. It works backwards to how you would expect. So in the vertical area, adding makes you go up, subtracting makes you go down. In the horizontal zone, though, on the other hand, adding makes you go to the left, which is the negative x direction and subtracting actually makes you go to the right, which is the positive x direction. So things work a little bit backwards when it comes to the horizontal zone, the horizontal, what I write domain there, the horizontal zone. See, that's the things in the horizontal zone. Things work weird. We don't even know how to write words correctly. Clearly, what I wrote there is you look inside there to kind of help you figure out the domain of things. But let's switch on to the next one. The next type of transformation we're going to talk about is the idea of scaling. And I don't mean like scaling a fish. I mean about changing the scale of a graph. This is often called compressing or stretching. Compression typically means the scale got smaller. Stretching typically means the scale gets bigger. And how does one detect a vertical scale? A vertical stretch or compression. Well, shifting happened by adding or subtracting a number to the x or y coordinate. Stretching and compressing is going to come about by multiplying or its inverse operation of dividing, dividing the function, either it's x or y coordinate by some quantity. Again, in all of these situations, the number and play here, we're going to consider positive. If you're interested in negatives, we'll talk about those in just a second with reflections. But if we have a positive number, if you stick a positive front of the f of x, this is going to affect the y coordinate and you're going to end up getting a vertical, a vertical stretch or compression, right? The vertical stretching happens when a is bigger than one. So if you times the y coordinate by something bigger than one, it's going to get bigger. And this has the effect of a vertical stretch on the graph. On the other hand, if your number a is a small number, so it's between zero and one. Remember, we're not allowing for negatives at the moment, but if we pick a numbers less than one, this is like dividing by a big number, or you think of multiplying by the reciprocal, we'd be like, oh, I'm multiplying by one third. I'm multiplying by one half. I'm multiplying by one fourth. This has the effect of shrinking the vertical coordinate, aka the y coordinate on the graph. And we refer to this as a vertical compression. So if you multiply f of x by a big number, it gets bigger. If you multiply by a small number, it gets smaller. Represent is either vertical stretches or vertical compressions, like we mentioned before. And so I often denote such a thing as you draw a little double arrow going up and down, because this kind of suggests like the graph is being stretched by a factor of a. And if a, of course, is one half, then you stretch something by one half is actually a compressed by two. You can think of it that way. Well, in the horizontal zone, things work a little bit differently here. If you were to replace, so with the vertical stretch you to compress, you just multiply the entire function expression by a to do the equivalent thing for the horizontal zone. You have to replace each X in the formula with a one over BX or just X over B if you prefer X over B. So you're going to replace each X with X over B in the formula. What's going to happen? Why are we dividing by B? Why not multiplying? Well, in the horizontal zone, things work not the way you would expect that in order to stretch the graph horizontally. You actually are going to divide by B in contrast. If you want to compress the function horizontally, you're going to divide by a small number. So you would divide by like, say, one half that would compress it, but dividing by one half is the same thing as multiplying by two. So it's going to feel a little bit backwards to what you expect. And we'll show you examples of this in just one moment on on a calculator on a graphing calculator. I just wanted to go through all of the transformations first. And then the last family of transformations we're going to talk about is reflections. There's only two type of reflections to worry about here. If you stick a negative sign in front of F of X, that's the same thing as multiplying the function by negative one. This will have the effect that you reflect across the X axis. And so reflecting across the X axis, you have your X axis right here as your horizontal line, and you're going to reflect across it. That's a reflection across the X axis there. On the other hand, if you want to do a horizontal reflection, you were you were going to replace X with a negative X, and this will have the effect of reflecting across the Y axis. So in that situation, your Y axis runs vertically. Get your Y axis right here and you're going to reflect to the other side. And again, we'll do some examples of all of these things in just a moment. And so this then summarizes the different types of transformations we can do. We can do shifting, we can do scaling, and we can do reflections so we could we could scale vertically or compress vertically horizontally by a factor of A or B. We could reflect across the X or Y axis or like what we saw on the previous screen, we could shift things up, down, left, right. And we can also combine all of these operations together. The reason that compressing and scaling required A and B, B positive is that if it's negative, you can just combine the stretch with a reflection and we'll play around with these with the following examples. What I want to next mention here is that in order to, in order to kind of effectively show you how to do this, I want to show you how to do this with a graphing calculator, some type of graphing utility. If you have access to a graphing calculator, it can be very useful in this situation, but be aware, I do expect students to be able to learn how to do these graph transformations without the use of a calculator. This is just something that we're going to see here just for the sake of the lecture. And what I'm planning to use here is actually one of my favorite graphing utilities, which can be found at www.desmos.com. Let me write that out for you www.desmos.com. You can also find the link in the description of this video here. And so Desmos actually offers a free online graphing utility. So if you don't have a graphing calculator, you're going to want one for this class, but I wouldn't go out and buy one at the store. Just use Desmos. It's completely free and looks like the following here. And so what you have and what you see in front of you now is is a graph from Desmos.com. I have currently graphed the function f of X equals X cubed, which you can see on the screen. It looks like it's an orange right now. And I've listed two important points on the graph. There's this there's this point zero zero, which you can see right here. And there's the point one one, which are both points on the function f of X equals X cubed. Don't worry about the function X cubed two months. It's just it's a good toy to play with right now. And so what we're going to do is I'm going to start showing you how these transformations affect this graph here. And so what we're going to see happening is the following. Let's see. I lost my mouse. Give me one second to bring it back up. That's not what I wanted to do. Here we go. All right. So you have the screen here again. So what we're going to do is if you do all the possible combinations, transformations to a function, you get something like the following. You could get Y equals a times f f of X minus H over B plus K. And this is if you combine all the things together because I want to mention that all of the transformations we're going to see here are just a result of function composition. And I'm going to show you what that means in just a second. I want to manipulate the graph right now. So what happens here is if we start moving our K value, right? So K, K, which you can see right here, K affects the vertical shift. So making K get bigger makes the graph go up. Making K get smaller makes the graph go down. And so, for example, you see here as K gets bigger, it moves the graph up, up, up, up, up. As K gets smaller, that makes the graph go down, down, down, down, down, down. So you can move it up, move it down. These are going to be our vertical, vertical shifts. I'm going to set K back equal to zero. If you move, if you change H on the other hand, H was this number. Remember, we see inside of, inside of the function right there. If you have X minus H, then the H, if we allow H to get bigger, it actually moves the graph to the right. And you actually have access to this page here on Desmos. You can find it in the link on the lecture notes or in the lecture videos that you have playing right now. You can play around with this yourself if you want to. In which case, making H become negative. So if you're subtracting a negative, you're actually adding a positive. This moves the graph to the left. And so just like we predicted, a subtracting H makes it go to the right and adding H actually makes it go to the left. So in the horizontal zone, things work backwards than what you might expect. If we change this value, A, this was a vertical shift, right, a vertical stretches, excuse me. As A gets bigger, it vertically stretches the graph. And so you can see the graph looks like it's getting taller, taller, taller. If we allow A to get smaller, right, it gets something less than one. You see that it's getting vertically compressed. It's like we throw the shredder into the trash compact or the back of that garbage truck. It's getting squished. And if you take a negative value, things got reflected. For example, if you look at A equals negative one, what happened was our graph is being reflected across the X axis, like you can see right there. And if you take more negative, combine the things together, you can vertically stretch it as well. Put it back to one. The default value would be A equals one. For B, on the other hand, if B gets bigger, notice the graph is getting vertically stretched. As B gets bigger, we stretch it, stretch it, stretch it. Now B is the value we're dividing inside of the function. On the other hand, if B gets smaller, you're going to get a horizontal compression. So it's getting skinnier, skinnier. If you take a negative value, it reflects across the Y axis in this situation. And then we can, of course, put all of these things together, right? I've actually set this thing up so you can manipulate it. So you can actually move the vertex, this point at the middle, where the origin ones, you can move it around. If you want to stretch it, you can do that, reflect it. And so this graph right here in real time, you can manipulate the graph. And this is what we meant when we started off this video. We can change the graph so I can stretch it, reflect it. I can move it, shifting it, shift it up, shift it down, shift it right, shift it left. You can do all of these graph transformations to the graph, but this also changes the formula here. Every time I adapt one of these parameters, A, H, B, and K, here, all of these have an effect on the formula, but it also changes the graph. So the two things are related to each other. Coming back to our list here, and this list here, this table, then enumerates all the possibilities that one could see. The different types of reflections, shifting in and scaling of some kind. So just the last thing I want to mention here is that we should represent that these different transformations are just function composition of some kind. So if you start off with like a function y equals f of x, if you want to do some type of like let's shift it, let's shift it right. By a factor, let's say by two, what you're going to do to shift it right, you then take y equals f of x minus two, you replace the x with x minus two. But I want to mention that we can decompose this function in terms of function composition. We can write this as f of u composed with x minus two. That is in terms of function decomposition, we can factor this in which case shifting to the right, so shifting to the right by two is just like, oh, I'm just going to I'm going to shift it right, x minus two, and then compose that with the function. On the other hand, let's say that we wanted to we want to do a little bit more complicated, we're going to we're going to stretch. We want to stretch vertically by a factor of two, and we're going to then shift and the shift up by a factor, let's say of three this time. What happens with your function? You start off with y equals f of x. Well, the first the first if you do this in stages, right? The first thing I said is we're going to stretch it by a factor of two. So we're going to take two times f of x. And then if you want to shift it up by three, you're going to shift it up by three like that. And I want to mention that this can be decomposed into function composition again. So the fact that we added three here, you're going to get this x plus three. You're going to compose that with, say, the next function, which is going to be two times x. And then you do one more compose that with f of x here. And so what you then see here is that you can decompose each of these transformations based upon one of these operations. And so what you see is that the horizontal zone is just the stuff that we apply before we get to the function and the vertical transformations are just those operations we apply after we do the function. So x comes first, then y when it comes to this function. That's because x is the input, y is the output. And so by recognizing function decomposition of these formulas, we'll be able to use this to help us identify correctly different graph transformations. And so this is a bit of a lengthy introduction. Following this video, you're going to see lots of short videos of specific examples of doing graph transformations with specific formulas and graphs. So take a look at those, take a look at those videos right now.