If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. In this unit we review the basic concept of a function and emphasize multiple representations of these foundational tools. Graphical features of functions, including maximums, minimums, intervals of increase and decrease along with domain and range are introduced. Classic function notation is used throughout the unit. Average rate of change is introduced as a tool for measuring the growth or decline in a function. We hope that visitors will use these lessons and give us feedback to make them better.
Get Access to Additional eMath Resources Register and become a verified teacher for greater access. Already have an account? Log in Math Hello and welcome to another common core algebra one lesson. By E math instruction. My name is Kirk weiler, and today we're going to be doing unit three lesson number four. On the graphical features of functions. Before we begin this lesson, 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. Also, don't forget about those great QR codes that are at the top of every single worksheet. Use your smartphone or a tablet. Scan that code and it'll bring you right to this video. All right, let's begin. The graphs of functions have lots and lots of important features, with lots of terminology that you're expected to know. All right? So we're going to get into some of that terminology today. Some of it will be reviewed from previous lessons that we've had. So let's jump right into it and exercise number one. Now it says the function Y equals F of X is shown graphed below over the interval negative 7 is less than or equal to X, which is less than or equal to 7. Now just as a side note, when somebody gives you this sort of like notation. What that tells you is that is what the function is defined over. Okay? Those are the values of X that we can put into the function, and we've got nothing outside of negative 7 to 7. So let's take a look at letter a letter asks us to find the maximum and minimum values of the function and a state the X values where they occur as well. All right? We've talked about maximum and minimum values of functions. So what I'd like you to do is pause the video right now. And see if you can figure out what these things are. The maximum and the minimum values of the function. All right, let's go through them. Remember, every time we talk about values of the function, what we're talking about is outputs, outputs, which are the same as the Y values. All right? So the maximum Y value occurs right up here. And the minimum Y value occurs right down here. So I guess we'll start with the maximum because that's the way the problem was worded. The maximum Y value is Y equals one, two, three, four, 5, 6, 7, at X equals. Negative one. All right. On the other hand, the minimum or the lowest Y value is Y equals negative one at. One, two, three, four, 5, X equals negative 5. That doesn't quite look like an X, so I'm going to just get rid of it there. There we go. All right, so whenever we talk about the values of the function, the values of the function we're talking about Y values. Now, something that you should have heard before, especially in terms of lines, is the Y intercept of the function, or the Y intercept. Letter B, what is the Y intercept of the function? And then it asks us to explain why a function can not have more than one Y intercept. Well, the Y intercept is simply the Y value where we cross the Y axis. I'm going to color it in in red, okay? Then maybe I'll circle this in red as well. Oh, we almost get purple. So the Y intercept is Y equals one, two, three, four, 5, 6. All right. Then it says explain why a function can not have more than one Y intercept. Ah, that's very, very important. Okay? Pause the video for a second and think about why a function, not just a generic graph. You can have a generic graph that has more than one Y intercept. But why the graph of a function can not have more than one Y intercept. All right, let's talk about it. All right, all capitals. All Y intercepts happen. When X equals zero. That's important. Every time you cross the Y intercept, it happens at a place where X is equal to zero. So if we had, let's say, two, Y, and I'm going to just abbreviate it that way. If we had two Y intercepts, then there would be two outputs. For that doesn't look like four. Let's try that again. For an input. Input of X equals zero. And as we know, at this point, after quite a few lessons on functions, for every input, there can be only one output. So if there were two or more Y intercepts, then there would be more than one output for an input of zero. All right, so only one Y intercept. On the other hand, let's flip that over, and let's take a look at C give the X intercepts of the function. These are also known as the zeros of the function because they are where F of X equals zero. So this is actually going to become an important piece of terminology for us. The zeros of a function. The zeros of a function. The zeros in the function are the X intercepts and their where Y is equal to zero, right? Let me circle those in green. They're right here. And write here, okay? The X intercepts are where Y is equal to zero, and that occurs at X equals one, two, three, four, 5, 6, at X equals negative 6, and at X equals one, two, three, four, negative four. All right. The zeros of a function are where the output of the function is equal to zero. The X values where the output is equal to zero. It's kind of cool. But Y intercept is where the input is equal to zero. The X intercepts are where the output is equal to zero. And that's why they're called the zeros in the function. Now, no problem with more than one, right? We can have outputs repeat. We can't have inputs repeat for functions. All right, there's a lot written down on this page with a lot of different colors. So pause the video now if you need to, because then I'm going to clear out the text. Okay, here it goes. Let's keep solving problems. Continuing with this one particular function. All right. Take a look at what it asks us to do in letter D would you characterize the interval X greater than or equal to negative 5 less than or equal to negative one as increasing or decreasing. Explain your choice. So let's go to X equals negative 5. One, two, three, four, 5. Here's negative 5. And here's negative one. So the question is, what's the function doing on this interval? Is it increasing or is it decreasing? All right, now this is important. Very, very important. Okay? Is the function increasing or decreasing? Now, for a lot of students, they would say, well, I mean, it depends on which way you're looking at it. If we move this direction, then the function is increasing. And if we move this direction, the function is decreasing. Right? So we have to understand how to read a graph. Always always read a graph. From left. To right. All right, now what do we do then? Why do we always read a graph from left to right? And it's very simple. When we read a graph from left to right, then that guarantees that our X values are getting larger. Because implicit in this question really is, as the X values increase, does the function increase or does it decrease? That's what it's always about. Is the function increasing? Or is the function decreasing? And if we now get rid of this right to left motion, because that's not correct. And we go back to always always always reading it this way, then what we see is that the function increases. So the correct answer is increasing. Because Y gets larger. As X gets larger. We go uphill. It's a good way to visualize it. As we're moving from left to right, we're moving uphill. Okay? Letter E says give actually it says given, but it should just say give. Give one additional interval over which the function is increasing, and one over which it is decreasing. All right? So to give an interval over which the function is increasing what we want to do is look for another portion where the function is going uphill. Okay? And a good example of that is from here to here. Right? So from two, all the way to three, four, 5. This is X equals two, X equals 5. From two to 5, we're traveling uphill. You could also use interval notation really nicely, right? You could go two to 5. All right? Likewise, if we're talking about decreasing, then we're looking for a place that we're going downhill. Why don't we why don't we use this stretch in here? Why don't we use this stretch in here to go downhill? We'll even keep our text and red. So it looks like we're going downhill from negative one, less than or equal to X less than or equal to two. Or interval notation. Negative one to two brackets. Whether you use brackets or parentheses there is mildly unimportant. All right? Those endpoints, it's kind of like the bottom of the hill and the top of the hill. All right? If you study calculus later on, which I hope that you all do because it's a great topic. We can either include them or not include them, although in calculus they tend to be included. Okay? So, you want to clear out the text, pause the video if you need to. All right, let's keep moving. Okay. Keeping with the same function over and over again. Something else some functions have are one known as turning points. Now our function has one, two, three, four turning points. And I think the name says it all. Turning points are also known as vertices, a single turning point could be known as a vertex. All right? But right now we're just calling them turning points. They are the top and the bottom of hills. If you are. There are also known as relative maximums and relative minimums. The reason that we use the word relative is because they are sort of the highest and the lowest points. But only in a certain area. You know, I live in upstate New York and we have the catskills nearby, and we can talk about sort of the highest and the lowest points of the catskills, right? And those are the relative highest and lowest points around here. On the other hand, there's certainly not the absolute maximum and absolute minimum points. Those might occur in the Alps or the Andes or other places. I'm probably way off in my geography. I'm a math teacher. Anyway, let's look at each one of these points. Here's our point at negative 5 comma negative one. Here's our point at negative one 7. Here's a point at two two. And here's a point at 5 comma 5. Which what are they relative maximums or relative minimums? Circle your choices, pause the video now, circle some choices and I bet you're going to get them right. All right. Well, negative 5 negative one is at the bottom of the hill, right? It's the valley, so it's a minimum, a relative minimum. Negative one comma 7, well that's at the top of the hill. So it's got to be a relative maximum. Two comma two, bottom of the hill, relative minimum, 5 comma 5, top of the hill, relative maximum. Not surprising that the alternate back and forth. It's probably hard to even imagine how you could have a graph that doesn't have them alternate, right? Relative mass relative min relative max relative min relative max relative men, et cetera. There are possibilities. But they require a graph to become what's known as discontinuous. All right, and we're not going there. Not quite yet. I think that actually completes exercise number one. I'm going to clear this out, so write down what you need to, and then we're going to move on. All right, all clear. Let's keep going. Back to our piecewise functions. Okay, fair enough. Remember, piecewise functions, and this one is known as a piecewise linear function because both of these are equations of lines. We'll get into those more later on in the course. But piecewise linear functions are simply rules that are given or that are created by combining two or more smaller rules, if you will. Okay? Remember the way that we interpret this. And I'll go with different colors here. Let's start off with red is that I'm going to use the formula X plus three any time X is less than or equal to one. So wow, that's like all of these, right? And then I'm going to use the rule 6 minus two X any time X is greater than or equal to one. That's weird that I included one in both of them. Maybe that's a typo. Or maybe it's not. Anyway, we'll put a little star here. And we'll come back to that. So let's start doing some calculation. Okay. So we got to use X plus three any time X is less than or equal to one, so we'll get negative four plus three. That'll be negative one, and we'll have the coordinate point negative four negative one. Likewise, when I have negative three, I'll get negative three plus three, which is zero, giving me the coordinate point negative three zero. Negative two plus three gives me positive one, giving me the point negative two comma one. Negative one plus three gives me positive two. That gives me negative one comma two. Zero plus three gives me three zero comma three. So let's use it anyway. And it says it's valid. One plus three, which is four. So I'm going to leave a little bit of room. One comma four. Okay. Now the other rule kicks in. If you don't mind, I'm going to kind of work this rule backwards because I want to leave that X equals one for last. All right? So remember, here we're going to do 6 minus two times the input. Now remember that multiplication has to come first, right? So that's going to be 6 -8. And that's going to be negative two. So we have the .4 comma negative two. Then we'll have 6 minus two times three. Again, the multiplication has to come first. So two times three is 6, 6 -6 is zero. Then we'll have 6 minus two times two. Which is 6 minus four, which is two. And let's see what happens when we apply that rule when X is one. Here we'll have 6 minus two times one. Which is 6 minus two, which is four. So we'll get the .1 comma four. Look at that. Same point. Now the reason that I did this in this exercise because again, at the cutoff point, those two outputs are the same, okay? So there's nothing wrong with that. What would have been a problem? What would have been a huge mathematical mistake on my part, the person who's designed this worksheet is if those two Y values came out to be different. If one of them had been four, and one of them had been 5, we would have had a serious problem on our hands. That wouldn't have been correct. All right, so let's do some graphing. All right, negative four comma negative one. I'll go in red. Negative three comma zero. Negative two comma one, negative one comment, oops, two, zero, comma three, one comma four. I could have put that one in blue as well. I think I'll switch to blue. One comma four, we already have, again, same point, two comma two, three comma zero, and four comma negative two. All right, let me connect them up. I'm already in blue and red. So I thought I was in red. Problem is I hadn't switched over to my line applicant application. All right. And maybe a little arrow. Because it'll go forever that direction. And let me go here. And again, nice little arrow. And there's our piecewise function. Love it. Right? Now, what we're going to do though is we're going to move on. You're going to have that table in that graph. And what I'm going to do is I'm going to reproduce the graph force so that we can then talk about the graphical features of this particular function as we work through the problem. So I have to clear this out just because of the way my software works. So write down anything you need to. All right, here we go. All cleared. Let's keep going. Now what I've done is I've put the graph up here. I did it all in red. That's okay. Let her be asked me to state the zeros in the function. Now we introduce that idea previously, see if you remember what that means, or look back at the front side of the sheet. Pause the video now and see if you can answer both B and C. All right, let's go through them. Remember, the zeros of the function are the places where Y is equal to zero. That's right here. And right here. That happens to be X equals one, two, three, negative three. And X equals one, two, three, positive three. All right? There are zeros. Functions Y intercept, that isn't too bad. Wow. I have a lot of threes here. Don't I? Big fan of three in this problem. Y equals three. All right? Now let's talk about increasing decreasing. This could be confusing, especially because of the arrows we've put on our graph. But remember, even though I might have this arrow here in this arrow here, I will always read the graph in this direction. So it says give the interval over which the function is increasing, give the interval over which the function is decreasing. Well, the function is going to be increasing all along here, right? So in fact, it's going to increase from negative infinity, can't put an equal sign there. Less than or equal to one. Whether you put that equal there or not, it doesn't really matter. In interval notation, negative infinity to one all kind of keep the bracket there. It's decreasing from one out to infinity. Again, whether you have the equals there or strictly a greater than less than simple doesn't matter. An interval notation one to infinity. Love the interval notation for these things. Consider using that, right? Where you start where you stop brackets parentheses. What are we? Give the coordinates of the one turning point and classify it as either a relative maximum, or a relative minimum. Why don't you go ahead and pause the video and see if you can answer E on your own. All right, let's go through it. Well, there is only one turning point. Let me mark it in green. Actually, let me erase some of this. All right, now we'll go back to green. There's my one turning point. It's got coordinates of one, one, two, three, four, so one comma four, and that is a relative. Maximum. All right, letter F, use your graph to find all solutions to the equation F of X equals two. Illustrate your solution graphically and find evidence in the table you created. A table which we unfortunately don't have the screen on anymore. And would be hard to have the screen. But you've got the table on your own paper. So hopefully you can look at that, or you could, of course, rewind the video if you need to. Now, remember, here, what we're given is we're given an output. Sorry that this is all the way on the bottom of the screen. But we're given an output of Y equals two and what we really want to find is what the X values are. Okay. So you know what I'm going to do. I'm going to actually draw in a nice black line right at Y equals two. Well, or at least I'm going to try to make it a Y equals two. That was a little bit too high for it. Sorry about that. Now look at that line. All right. Right here is one place where the output is two, and here's another place the output is two. So solving that equation gives me X equals negative one and X equals positive two. All right. So that's tricky. When given an output, when we're trying to find inputs, that's a much harder task, a much, much harder task. Finally, letter G asks us to state the interval over which this function is positive. How can you tell this quickly from the graph? All right, now I know that there's a lot kind of drawn up on that graph right now. But I need to erase it because I need to be able to explain this important topic to you. All right, now notice what the thing says. It says state the interval over which this function is positive. I'm going to highlight this now. This function is positive. Okay? Very, very important. All right, the function is the Y values. Now, just think a little bit. Going to the graph. Here's a Y value that's positive. Right? But here's a Y value that's negative. Here's a Y value that's paused. Here's a Y value that's negative. In fact, all the Y values up here are positive. And all the Y values down here are negative. So the place where the function is actually blue is actually positive. Is everywhere in here. Everywhere where it's above the X axis. This is where it's positive. It starts being positive, not at negative three, but very close to negative three. Right? And then it ends being positive at X equals positive three. Now we can not include the equals there. We can not include the equals there. Because it's not above the X axis. It's at the X axis. Those are its zeros. But we can not include the negative three and the three because it's not positive there. Interval notation here, I think, is great. Negative three to three. Of course, the most problem part of interval notation. Is the fact that this kind of looks like a coordinate point. But it's not. It just means that the function is positive, starting at this value, ending at that value, but not including them. Okay? Very important. We can always tell that a function is positive, get that out of my way. We can always tell that a function is positive, where it's above the X axis, and negative where it's below. But what's interesting is that we give the X values that give us positive Y values. And we give X values that also give us negative Y values if we're asked where the function is negative. All right, a lot on this page. I'm going to clear it out. So pause the video now and write down anything you need to. All right, let's go on. Let's finish up the lesson. Today we introduced a lot of important function terminology. Some of which are things like the Y intercept, the X intercepts, also known as the zeros of the function, all right? We also introduced the idea of a function increasing over particular interval of X, a function decreasing over a particular interval of X, right? We talked about relative maximums minimums and how they relate to this idea of a turning point. And finally, we also saw where functions were positive and where functions were negative. So we're going to be using that terminology a lot in the future. Make sure that you watch this video or talk to your teacher as much as you need to to understand these terms. Thank you, though. For joining me for another common core algebra one lesson by email instruction. My name is Kurt weiler, and until next time, keep thinking. And keep solving problems. Remove Ads |
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