Common Core Algebra I.Unit 11.Lesson 1.Function Transformations

Hello and welcome to another common core algebra one lesson by E math instruction. My name is Kirk weiler, and today we'll be doing unit 11 lesson number one on function transformations. Before we begin, let me remind you that you can find a worksheet and a homework assignment that go along with this video by clicking on the video's description. Or by visiting our website at WWW dot E math instruction dot com. As well, don't forget about the QR codes on the top of every worksheet. You can use your smartphone or a tablet to scan that code and it will bring you directly to the video associated with that lesson. All right, let's begin unit number 11. So you know 11 is a real grab bag of topics. You know, high level topics that we wanted to see one more time. And one of the biggest ones is the idea of taking a function and transforming it. That means shifting it as we've seen with parabolas and square root functions and absolute value functions. But today, we're going to look a little bit more abstractly at the idea of stretching functions. All right? So let's jump right into exercise one. Something that you should really be able to do at this point. In exercise one, we're given the function Y equals F of X and letter a asks us to evaluate the function at a bunch of different inputs. So I'd like you to pause the video right now and see if you can do that, okay? All right, let's go through it. So remember when we were giving something like F of three, that three is the X value. And we're giving the Y value. So it's really easy. We just go over to three, right? And we go up and we see that the Y value is two. When we're asked for F of negative four, the same idea, we go over to X equals negative four, we go up, and we also have a Y value of two. F of 7, well that's three, four, 5, 6, 7, and we go down and we have a Y value of negative two. And F of negative 7, one, two, three, four, 5, 6, negative 7, one, two, three, four, gives us a Y value of negative four. All right, that's not so bad. How about letter B? State this zeros of F of X pause the video now and see if you can do this. All right. Let me change to a different color. Blue. Remember, the zeros are where Y is equal to zero. Well, that occurs here. At X equals one, two, three, four, 5, negative 5. And it occurs right here at one, two, three, four, 5. X equals positive 5. Now letter C is quite important. It says, why is it impossible to evaluate F of 9? Pause the video now and think about that for a minute. All right, well, hey, if it says it's impossible to do it, I say we dare them and try to do it anyway. Let's try. One, two, three, four, 5, 6, 7, 8, 9. Here's 9. And the problem is the function doesn't exist there. So there's lots of different great explanations you could give for letter C you could say the function doesn't exist when X is 9. You could say the graph doesn't extend to X equals 9. Or X equals 9 is not in its domain. The domain is the set of X values that will give us a Y value and 9 isn't in it. In fact, that's what letter D asks is for us to state the domain and the range, remember the domain are the input values, and the range is the set of output values. So what's the smallest X value? Well, it's negative 7, and the largest is positive 7. So one good way of stating the domain is like this. But if you remember interval notation, I like that as well. That's a big fat or you certainly don't need both of them. Now the range is a little bit trickier. Let's figure out our minimum Y value one, two, three, four, minimum Y value is negative four. Our maximum Y value is two, so there we go. Negative four is less than or equal to Y less than or equal to two. Or if like me, you like interval notation, I'd say negative four to two, and remember those brackets mean that they're included the negative four. And the two. All right, so just a little warmup problem. Nothing to do with function transformation right now. Until we get into the next exercise. So I'm going to clear this out, pause the video now if you need to. Okay, here we go. Exercise two. It says now, let's now define the function G of X by the formula whoopsie got cut off a little bit there. G of X equals two times F of X all right, now remember, this is still the graph of F of X this is not G of X but letter a says evaluate each of the following. Show the work that leads to your answer, remember just follow just following the function's rule. How about just follow the function's rule, okay? So simple enough G of negative 7 if we follow the function's rule would be two times F of negative 7, right? Wherever there's an X, we put in an X so G of negative 7 is two times F of negative 7. Well, what was F of negative 712-345-6712, three, four. Well, F of negative 7 was negative four. So G of negative 7 is negative 8. G of negative four will be twice F of negative four. Well, what was F of negative four? One, two, three, four. Oh, it was positive two. So G of negative four is positive four. G of three, well, that will be two times F of three. Well, F of three was one, two, three was two. So that will also be four. And G of 7 will be two times F of 7, and F of 7 was one, two, three, four, 5, 6, 7, F of 7 was negative two. So G of 7 will be negative four. Now, this really leads us to letter B how can you interpret the function rule in terms of the graph of F of X? Well, think about what we did letter a every one of those Y values simply got multiplied by two. So let me word it this way. Each Y value. On F of X. Will be. Multiplied by two, to get G of X that's it. Each Y value on F of X will be multiplied by two to get G of X letter C says sketch the graph of F of X on the grid above and exercise one. I mean, we can't go back to that. That's why I put the grid there again. Write down the points that you know are on G of X based on your answers to a so in other words, like take a look at a in this problem, right? When the input was negative 7, the output was negative 8. Let's do that one. 123-456-7123, four, 5, 6, 7, 8. In other words, this point got transformed down to here. Then let's go with the next one. When we had negative four, our output was four. So one, two, three, four, one, two, three, four. So this point went up to there. All right. The next one, we had the .3, and it went to four, so one, two, three, one, two, three, four. Again, this point went up to here. And then we have the .7 comma negative four, 123-456-7123, four. This one went down to here. I'm going to now draw this in red. I think I'll use the oh, it's already on red. I'll use this to connect them. So that there's that one, not the best, and here's that. Look at that. And that is the graph of G of X. In red. It's a state the domain and range of G of X well, um. It looks like G of X still starts at negative 7 on X and still ends at 7. As well on X so notice that was the same as F, right? That didn't change. But the range is going to change. Look at this. The lowest Y value is down here at negative 8. And the highest Y value is now up here at four. So our range is now negative 8 to four. Let me put this in non interval notation in case you don't use interval notation in your class. And notice all that range was was if you will, twice the range of the original function. That's not a coincidence, right? It's all about that too. Okay. So pause the video now and copy down anything you need to. All right, here we go. Clear in and out. Wow, all gone. So let's summarize this. Okay? What we just saw was what's known as a vertical stretch. And vertical stretches occur when we take an original function, and we multiply by some constant, okay? Now, that constant could be bigger than one in its absolute value. So let's not worry about negatives right now. So the function is going to get stretched like it did in the last problem if our constant is bigger than one. So like if you have a K value of two. On the other hand, you could actually compress the function if your K value is less than one. So if we had a one half then all the Y values would be just one half as high. But there is that third effect, which is what happens when our K value is negative. Well, that will not only stretch or compress the function, but it will also reflect it across the X axis. Okay? All of this, we've seen before, specifically in the context of parabolas. So one more time, right? If we multiply by a number greater than one, the parabola or anything else gets stretched, right? If we multiply by a number that's less than one, it gets compressed, and if we multiply by a number that's negative, it'll both get stretched or compressed, and it'll flip across the X axis. All right, I'm going to clear out that little bit of text. And then let's see another example. All right, exercise three. A quadratic F of X is shown below. We don't even have its formula. The function G of X is defined by negative one half times F of X, looks like really see this. Negative one half times F of X but a race is calculate values of G of zero and G of three show your work and then explain the effect of multiplying by negative one half. We'll watch. One of the things I love about function notation is that it's very straightforward. Anywhere there's an X in this formula, we're going to replace by zero. So we have negative one half times F of zero. Now, of course, you have to ask yourself, well, what is F of zero? You come over here and F of zero is negative two. So negative one half times negative two is positive one. Likewise, G of three will be negative one half times F of three, but now I have to figure out what F of three is. Let's see, one, two, three, one, two, three, four. And negative one half times positive four is negative two. So what happened, right? We took this point and it's going to end up actually going up to this point. On the other hand, we also took this point, and it's going to end up going down to this point. Okay? So we did two things. We compressed the graph. And we'll see this once we graph it. We can press the graph, and flipped it, let's use that term for now. Flipped it across the X axis. All right. Now it says sketch an accurate graph of G of X on the same grid as F of X well, you know what's kind of cool about this is that if you really get this and actually I'll get rid of all of it right now. It's not that bad. Because what you always do is you take a Y value like this one, two, three, four, and you just multiply it by negative one half. And then you plot that at the same X value. Then I could take this Y value at negative two and multiply by negative one half. And I'll get positive one. Is that then goes there? This Y value, which is what, down there at negative four, multiply it by negative one half, and we get positive two. This Y value was the same as this one, right? And it went up here. So this one's going to go up there. And that one was the same as that one. So it's going to come down here. And so we end up getting. That's a decent parabola for once, not great, but it's decent. Now, it may not be completely obvious that it's been compressed, but it has. I mean, think about it. This parabola is only two units above the X axis. Whereas this parabola is four units below it. But it should be clear that kind of flipped across the axis. Now, what's the range of that new function that we just drew? Pause the video now and write down the range. All right. Well, this is kind of a cool one, right? The maximum Y value now is two, and then it just keeps going down. So the easiest way actually to state the range here is Y is less than or equal to two. Or, if you wanted to get fancy with function notation, or sorry, interval notation, it goes from negative infinity to two, including the two and not including the negative infinity. All right. That's kind of neat. Pause the video and copy down whatever you need to. Okay, here it goes. So let's do one more problem just to make sure you have this. Okay? We've done problems like this before with parabolas, but now we're going to do them with a generic function. So in this problem, I've got the function F of X in that bold line. So it's that one. It's that bold one, right? And then what I've got is I've got three other functions, G of X is one half of F of X H of X is negative F of X and K of X is two times F of X and I want you to write which one is which in each of these blanks. So do this as a self assessment. Pause the video now and try to figure out which one is which. All right. Well, let's do it. I can just knock them off one at a time. See, G of X is one half times F of X that means if I had a Y value on F of X, the G of X should have one that is half as tall. Well, that's this one, right? That's half as tall, so this must be G of X now, H of X is negative F of X, right? Which really just means that and I'm gonna erase this, is that if I take this Y value, let's say this Y value was. I'm just going to make up something. Let's say it was two. Right? Then H of X should be at negative two. Oh, that's this one. Which means it's got to be that function. Now that leaves us only that one, but keep in mind that what this one says is if I take the Y value on F of X, the Y value on K of X must be twice as high, and that's got to be here. All right. And that's it. So pause the video now, and then I'm going to clear out the text. Okay, here it goes. So today, what we saw was a very specific type of function transformation, a vertical stretch slash compression. All right? And what we learned was if we took a function and we multiplied it by a number greater than one, it'll take every Y value on that function and multiply it by that number, stretching the function. On the other hand, if we multiply it by a number between zero and one, so let's keep the negatives out for right now. It will compress it, right? Finally, if we multiply it by a number that's negative. All right, assuming it's not negative one, like it's negative two, negative one half. It will both stretch or compress it and it'll flip it across the X axis. If we multiply it by negative one, it just reflects it across the X axis. All right. For now, I'd like to thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weiler, and until next time, keep thinking. I keep solving problems.