Common Core Algebra II.Unit 7.Lesson 4.Horizontal Stretching of Functions

So what I'd like you to do is pause the video now. And use your graphing calculator to graph this original function F of X all right. So graph this thing. Pause the video now, go ahead and do that. Okay. Well, since it's our classic absolute value graph, what I'm going to do is I'm going to use my straight line command. And it will look something like this. All right, let me label that F of X and of course I should be able to figure out its turning point by just looking at this sort of vertex form. Of this absolute value function. So it's got the point to comment negative three. This graph is going to get a little crowded. Letter B, consider the function G of X equals F of two X, determine a formula for G, then graph it on the axes. Use your calculator to find its minimum point and label it on the graph. Okay, fair enough. Well, let's find the formulas together because this is a little bit weird when you see something like this. All right, I'm going to actually do it in red because that's how my calculator is going to graph the next. The next function for me. So let me circle this in red. G of X is going to be F of two X and that's going to be two X minus two minus three. 

Remember all it means is that wherever there was an X, I'm going to now put in a two X so what I'd like you to do now is use your graphing calculator to graph that. I would actually keep F of X in there. And then use the minimum command that we learned about in the last lesson to find the minimum of that absolute value function. Pause the video now and go ahead and do that. All right. So let's do it. This is the graph that we end up seeing and ends up being something more like this. Now, that's not good. Let me get rid of that. And then of course I raised some blue. I'm going to try to put the blue back in there. There we go, change this back to red. Okay, it's a little bit hard because I want to make sure that we get certain things really, really accurate on here. So we end up getting kind of a function that looks more like that. There we go. I like that a lot better. And that's G of X and what we find is that it now has a turning point at one comma negative three. Okay. Now we're going to do exactly the same thing one more time. I'm going to do this in black. But we're now going to create the function H of X, which is F of one half X, all right? Same idea. This time I'd like you to create a formula for it, substituted into your graphing calculator, and then graph it. Okay. Well, let's come up with the formula first. 

So H of X is going to be F of one half X, which is simply going to be the absolute value. Of one half X minus two minus three. And when we graph that, we get something that looks kind of like this. And it barely this part barely fits on. I'm going to extend it a little bit past there, even though that wasn't in our window. What we know that's the case. And although I didn't do the greatest job sketching this, I apologize. It's turning point. Is it F isn't four negative three? So let's summarize. F are original function. Had a turning point at two negative three. Our next function, G of X, had a turning point. Of one common negative three, and finally, our function H of X had a turning point at four comma negative three. So letter E, what state constant about the turning points? Well, that was pretty easy. The Y coordinate. Y coordinate didn't change at all. It was negative three each time. But what did change? Well, the X coordinate. And it changed in a way that is a little bit counterintuitive, right? When we multiplied X by two, F of two X, the X coordinate was divided by two. And when we had F of one half X, DX coordinate was multiplied by two. Isn't that cool? 

Now, the only thing that's good about the counterintuitive nature of this, right? In other words, when you multiply X by two, it actually compresses the graph by a factor of two. And then when you multiply by one half, it stretches it by a factor of two. The only thing good about it is that it's a horizontal transformation, and it's kind of working like a horizontal shift in that it sort of works the opposite of the way you'd expect. You'd kind of think, well, if I do F of two X, then maybe every X value should be multiplied by two, but the crazy thing is every X value is divided by two. And when you have F of one half X, it's the same deal. Even though I've done one half times X, every value of X is actually divided by one half, or multiplied by two, depending on how you want to think about it. All right. There's a lot sitting here. So what I'd like you to do is pause the video now, write down anything you need to, and then we'll summarize this. Okay. 

Horizontal transformations of functions, whether they're shifts or stretches. Work the opposite of what you'd think. So take a look at this. Horizontal dilations for a real number, positive constant, such that K is greater than one. So again, K is positive. And it's bigger than one, then F of KX represents a horizontal compression of F of X by a factor of K and the function F of one over K times X represents a horizontal stretch of F of X by a factor of K all right. So again, very, very counterintuitive in terms of how this should work. We multiply X by a number bigger than one, and we end up dividing by that number. We multiply by X by a number less than one between zero and one, and we also end up dividing. I mean, if we multiply by one half, from multiply X by one half, then what happens is all those X values on the graph get multiplied by two. All right. It's kind of confusing. Let's jump into it and see how it works in the problems. Okay. Let's take a look at the quadratic function F of X equals X squared -12 X plus 20. Letter a determine the coordinates of its turning point by using the equation for the axis of symmetric axis of symmetry. Oh, lovely. 

The axis of symmetry of X equals negative B divided by two I so let's find the turning point of this using that formula we saw in a previous in a previous unit. Let's do it together actually. Been a little while since we've done this. Negative B will be negative negative 12 divided by two a, a is one. So we'll get 12 divided by two. So that's 6. So the X coordinate of the turning .6, and the Y coordinate, of course, we're just going to find by substituting the X coordinate into the formula. And we find that the Y coordinate is negative 16. All right, so there's our turning point of this guy. Now letter G, there are letter G, letter B if G is defined by G of X equals F of three X, what should the coordinates of its turning point based on our previous work? What should be the coordinates of its turning point based on our previous work? Explain. All right? Well, since we're multiplying X by three, this should be a horizontal compression. By three. All right. Now that means that the .6 comma negative 16 will go to the .6 divided by three, negative 16, so we should have a new turning point at two common negative 16. The Y coordinate won't change. Remember that last exercise with the absolute values that Y coordinate of the turning point didn't change, but the X coordinate kept doing it. 

Letter C says determine a value formula for G of X and then use the turning point formula to verify your answer from part B great. All right, well, what is G of X then? G of X is F of three X so that's going to be three X squared -12. Times three X plus 20. Three X times three X is 9 X squared. -36 X plus 20. So here's my formula for G of X. And let's find its turning point. X equals negative B divided by two a that's going to be negative negative 36. Divided by two times 9. That's going to give me a positive 36 divided by 18. And that's two. Awesome. Because that was two. And of course the Y I'm going to get by just doing G of two. By doing 9 times two squared -36 times two plus 20, and again, the remarkable thing about that is that it ends up being negative 16. All right, so what we see is by doing F of three X, our X value got divided by three, at least for the turning point, but the Y value didn't do anything at all. Now, letter D, which we can barely read because of all the arrows, let me get rid of some of those. Letter D. Says shows that the Y intercept for both F of X and G of X are equal. Why does this make sense from a horizontal dilation perspective? Okay. 

Well, Y intercept. That's simple enough. That's when X is zero. So F of zero is going to be zero squared -12 times zero, plus 20. Which is going to be 20. G of zero is going to be 9 times zero squared. -36 times zero, plus 20, which is 20. So yeah, they're the same. All right. And it makes sense because when you divide X equals zero by K. In this case three, it's still it's still X equals zero. Right? So the idea is if we really wanted to produce graph of G what we would do is we'd take every point on the graph of F and we'd take its X coordinate and we divide that X coordinate by three, but we wouldn't change the Y coordinate. There's that one point on the graph though, where X is zero. So when X is zero division by three, it's not going to do anything. Okay? Pause the video now and write down anything you need to. All right, let's get rid of this text and move on to the next problem. Actually, move on to the last problem. Exercise three, consider the graph of F of X graphed on the grid below. If G of X equals F of one half X for all values of X, then answer the following questions. Okay. 

So again, this exercise is going to give us a little bit more insight into why multiplying by one half is actually going to stretch this function horizontally. So let's evaluate some function values, right? G of negative 6, well, G of negative 6 will be F of one half times negative 6. That's just the rule. But that will be F of negative three, and what's F of negative three odds. One, two, three, four. So that means the point negative 6 positive four lies on the graph of G of negative two will be F of one half times negative two, which will be F of negative one, and that's this point down here. That's going to be negative two. So the graph negative the point negative two negative two also lies on this function. G of 6, well, that's going to be F of one half times 6. Which will be F of three. You probably already see where we're going on here, which is negative two. And therefore, 6 negative two lies on this graph. And G of 8 is F of one half times 8. Which is F of four, which is three. So the point a comma three lies on here. 

Graph G on the grid to the right. All right. Well, I've got 123-456-1234. Negative two, negative two. One, two, three, four, 5, 6, one, two, and 123-456-7812, three. Use my straight line command. And there it is. How would you describe this graph compared to the graph of F of X? It has been. Horizontally stretched. By a factor. Of two. Stretched or dilated. We took that whole graph, took every X coordinate on it, and just went like this. Multiplied every one of them by two. Every one of them. It's really cool, right? This point, right here, which was that negative whoops. Which was that negative three comma four suddenly became the point negative 6 comma four. This point down here, which was at negative one negative two. Became the point negative two negative two. Nothing happened to the Y coordinates. But the X coordinates kept getting multiplied by two. Isn't that cool? All right, gives you hopefully a little bit of a feeling for why this just like horizontal shifting is kind of counterintuitive. 

All right, pause the video now, write down anything you have to, then we'll wrap up the lesson. Okay, here we go. All right, so today we saw horizontal stretching of functions. And again, kind of like horizontal shifting, it worked the opposite that you'd think. If we have something like F of 5 X while it would mean that we take every X value on the graph of F and divide by 5. On the other hand, if we had F of one 7th times X, we would take every X value on the curve and multiply by 7, but you don't do anything to the Y values. All right. We'll play around a little bit on the homework with combining these things. But for now, I just want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler. And until next time, keep thinking. and keep solving problems.