Alg1.6 Introduction to Quadratic Functions
In this unit, students study quadratic functions systematically. They look at patterns which grow quadratically and contrast them with linear and exponential growth. Then they examine other quadratic relationships via tables, graphs, and equations, gaining appreciation for some of the special features of quadratic functions and the situations they represent. They analyze equivalent quadratic expressions and how these expressions help to reveal important behavior of the associated quadratic function and its graph. They gain an appreciation for the factored, standard, and vertex forms of a quadratic function and use these forms to solve problems.
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Lessons
A Different Kind of Change
Quadratic Functions
Working with Quadratic Expressions
Features of Graphs of Quadratic Functions
Math
Hello, I'm Kirk weiler and this is common core algebra two. By E math instruction. Today, we're starting a brand new unit. We're going to be doing unit 6, lesson number one on quadratic function review. Unit 6 is all about quadratics, primarily parabolas, but we're going to see a few circles towards the end. And even though you extensively covered quadratic functions in common core algebra one, we want to make sure that in this unit you see both the basics of quadratic functions, including their graphs parabolas. And that you also see their algebra, including extensive work with factoring in the zero product law. But in this lesson, we're just going to review some basics about quadratic functions. So let's begin. All right. First off, what's a quadratic function? It's any function that can be put into this form. AX squared plus BX plus C, where the coefficient on the X squared term isn't zero. All right. Now that's important because B and C can certainly be zero. In other words, you can certainly have Y equals X squared. That's a quadratic. Y equals 5 X squared certainly a quadratic. You could certainly have Y equals, let's say, negative two X squared minus three X, that's fine. But what you can't have is a being zero. Because if a is zero, if you had something like this, no X squared term, well, then it's no longer a quadratic, but it's a linear function. The equation of a line. So it's the inclusion of the X squared that takes it up a notch and gives us what's known as a quadratic function. All right, let's play around with these things. Exercise number one without the use of your calculator. Evaluate each of the following quadratic functions for the specified input values. Recall that, according to the formal order of operations, exponent evaluation should always come first. All right? How about this? How about you take this as a moment to make sure that you understand your order of operations and how to interpret things? Pause the video and be especially careful in letter C, but don't use your calculator. All right, go ahead. Okay. Well, let's take a look at letter a very, very simple. F of negative three, what's our rule say? It says take our input and square it, negative three squared is positive 9, right? This one, it's almost not even worth writing down. 5 squared is 25. Now when we have G of X, it's a little bit more complicated. If we're looking for G of two, then we absolutely must square this two first, which gives us four. And then we'll do two times four, which is 8 -5, gives me three. Likewise. This one, I'm going to put that negative one in first. I'm going to square it. I'm going to get one, positive one. Whenever we square a number, it's a positive unless it's zero. And then we just get zero. Here we'll get negative three. Okay? Now, this one is the harvest. Okay? And it's the hardest because a lot of students will do the following. They'll go, oh, I'm supposed to take negative X, so they'll take negative two, and they'll negate it, and they'll get two. And then they'll square it. No, that's wrong. The way we have to interpret this is very similar to how we interpreted letter B okay. Meaning that I really want to look at this as negative one X squared plus four X, in which case what I have is I've got negative one times negative two squared plus four times negative two. In which case I square that negative two first, I get positive four, but then I'm still going to have to multiply, I'm going to do that multiplication right away four times negative two is negative 8. But then I'm going to still have to multiply by the negative one. So I get negative four -8. Which is negative 12. If you got that negative 12, then you're doing well in terms of the way that you interpret this formula and order of operations. Let me do this one as well. Negative one times three squared. Plus four times three. Again, negative one times 9. Plus 12. Gives me negative 9 plus 12. Gives me a positive three. Apparently, I like the number three a lot in this exercise. All right, so just a little order of operations, making sure that we're ready for quadratic functions and all they bring. Pause the video now and write down anything you need to. All right, let's get rid of it. And let's move on. Okay, exercise two. Consider the simplest of all quadratic functions. Y equals X squared. Letter ace has created a table of values to plot this function over the domain interval negative three to three. All right? So go for it. Plot this. Okay, let's do it. I'm not going to use a calculator at all, right? I'm going to do negative three squared. I'm going to get 9. Negative two squared, I'm going to get four, negative one squared one, zero squared, zero, one, four, and 9. And so sketch a graph of this function on the grid to the right. Great. So I'll do negative 312-345-6789. And then I'll do negative two, one, two, three, four. Negative one, one. Zero, zero, one, one, two, two, and three 9. And now for the wyler guarantee of a badly drawn graph. Oh man. Now, little comment on arrows. Okay? It's very understandable when students put arrows on this because the parabola does continue to go up forever. On the other hand, it says create a table of values to plot the function over this domain interval. So if it's just this domain interval and only this domain interval, then that's all we should do and we shouldn't extend any arrows. Okay? No arrows on this. Arrows. So we've sketched the graph. Now it says state the coordinates of the turning point. What are the coordinates of the turning point? You should be able to write those down. All right, the turning point also known as the vertex of the parabola is probably the most important and identifiable point on the parabola. On the simplest of all parabolas, it's at the .00. The origin. All right. Now it's a state the equation for this parabola axis of symmetry. The axis of symmetry is a vertical line. I'm going to draw it in, I think I'll draw it in blue, a vertical line that cuts the parabola into symmetric portions. So its axis of symmetry is that blue line. And since it's a vertical line, it has the equation X equals the number. Remember how vertical lines have equations like X equals four, X equals negative two, et cetera. And specifically, this one has the equation X equals zero. Finally, it asks us to over what interval is this function increasing. Now the way I want to think about that is if I'm walking from left to right, always left to right. For what X values is the function going uphill. And there's really two acceptable answers here. We could say, well, it's for all values of X that are greater than zero, but less than or equal to three. In our problem. Or we could say that we include the zero. Either one of these answers is completely okay. All right? There seems to be a lot of confusion these days about whether or not with intervals of increase or decrease, whether you should include the endpoints like zero, specifically should you include turning points. And the official take on it is that there is no official take on it. That you can really do it either way, include the turning points don't include the turning points. It won't matter from an assessment perspective. And to me, I like it both ways. Do you include the top and the bottom of the hill? Or do you exclude the top and the bottom? I don't think it matters. Okay, pause the video now and write down anything you need to. All right, let's clear out the text. And keep working. Exercise three. Now we're going to get into our calculator tables a bit. Consider the quadratic function F of X equals negative X squared plus 6 X plus 5. And the first thing it asks us to do is to use a table on our graphing calculator to try to figure out the turning point of this function. Let's see how to do that. All right, I'm going to bring out the TI 84 plus. Now there's not a lot of room on the screen and I apologize because I'm going to have to like enlarge the calculator shrink. It move it, put it all over the place. My head's going to get in the way. It's going to be pure chaos. Cats hanging out with dogs. You know, things like that. Anyway, so let's put that formula into Y equals. Let's hit the Y equals button. Let's clear out any equations you might have in there. All right, and now let's put in negative X squared plus 6 X plus 5. As always, make sure that negative is the negative sign and not the subtraction sign. It'll give you an error if you put subtraction there, but then you might not know what the error is. Make sure your formula looks good. And then what's hop into the table. Now, what should we begin our table act? Well, that's a great question. Because I don't know what the turning point is. That's the point, right? I don't know where to look, things like that. But I will say that the problem is supplied me with graph paper. And the minimum X value on that graph paper is negative 6. So that's where I'm going to start my table. Is that negative 6? And I'm going to make my table go by ones. So let's set that up. All right, and let's now pop into the table. Okay. Now I'm going to bring my cursor over to the sort of Y column. And I'm going to start hitting the down arrow. And what I notice is I noticed that those Y values are getting larger. They're going up. Yeah, they're very negative, but they're definitely going up, right? They're getting closer to zero. Now, if I keep hitting the down button eventually, they become positive, right? And see, keep hitting them. Look at that, right? It's getting up to ten and 13 and 14. And then it goes back down to 13. You see how it changed direction and now it's starting to go down again. And also notice that symmetry. At X equals two, right? At two, we have a Y value of 13 at three, we have a Y value of 14, and at four, we have a Y value of 13 again. So that's how you can spot the turning point in a table. So the turning point has coordinates of three comma 14. It's going to be somewhere out here. All right. Letter B, what is the range of this quadratic the rage? The outputs or Y values. Well, all parabolas either look like this. And they have a range of Y less than or equal to something. Or they look like this. And then they have a range Y greater than or equal to something. This parabola looks like this one, because it has a leading coefficient that's negative. And I mean, we can even see that in the tables. That means it has a maximum, a maximum Y value, a max Y value, and that maximum Y value is right there. So our range is going to be Y less than or equal to 14. You can also put it in interval notation if you like that stuff. And it would look like that. I think I'm going to stick with Y is less than or equal to 14. Okay. Now it says graph this function. Graph this function of the grid to the right. Probably should say graph this function. On the grid to the right, that sounds a little bit better. So let's take a look at that table. I'm going to just start to plot points. Now, if I go up in the table until I hit sort of what I think will be the first point that lies up on there, that's going to be the point negative one, two, right? So negative one, negative two, sorry, not two, common negative two is going to be right here. Then zero, one, two, three, four, 5. Then we have one, 6, 7, 8, 9, ten, two, 11, 12, 13, three, 14, four, back down to 13, 5, set that ten more. 6 is back to the 5 mark. Look at all the beautiful symmetry and 7 is back there at negative two. All right, again, come on. This time I am going to put arrows on. Because they didn't say to graph it over a particular domain. They just said graph the function. So there it is. Not beautifully graphed, but there it is. Trust me, I would do a much better job if I was drawing on a piece of paper, but I'm drawing on a graphics tablet that's off to my side while trying to control my hand by looking at a screen. It doesn't do well. And never does well. But there it is, right? Now, we've kind of answered letter D, but let's make sure that we have it. Why does this parabola open downward? This is a downward opening parabola. Versus Y equals X squared, which we saw before that opened upward. Why is that? Think about it. Write something down. Pause the video. Now, what a lot of people will write down. And this is not a great answer. They're getting the idea right, but the answer is not great. It's a lot of students will say, this opens downward because the X that's being squared is negative. That's not what you want to say. With this formula really is, is it's really negative one times X squared plus 6 X plus 5. So it's all about that negative one. And the correct answer with proper terminology is the leading coefficient. The leading coefficient is negative. On Y equals X squared, the leading coefficient, the first coefficient is positive. It's a positive one. Don't say that the X squared is negative. X squared can never be negative. Whenever you square number, you get something that's positive or zero. So saying that the X squared is negative, would be a bad thing. All right, lastly. Between what two consecutive integers does the larger solution to this equation lie? Show this point on the graph. All right, so this is all about solving an equation graphically, right? And really what we're trying to interpret this as is find the X values where the Y value is zero. Well, the Y value is zero on this really poorly drawn graph right here and really somewhere right around here. I just can't draw the graph very well. This is the larger of the two. And it falls between one, two, three, four, 5, 6, and 7. Between 6 and 7. All right. Well, don't put that calculator away. We're going to need it for the next exercise. So I'm going to even leave mine out even though I may have made it quite small at this point. We'll make it larger in just a second. But pause the video now and write down anything you need to. All right, let's clear out that text. Oh, I almost snap my fingers, that would have gotten rid of the calculator. I'm going to want to do that. The magic snap. All right, let's take a look at exercise four. Again, we got a little stereo going on with the graph. My apologies for that. That'll be fixed in the workbook. A sketch of the quadratic function Y equals X squared -11 X -26 is shown below. Marked with its points that the intercepts and its turning point. Using tables or graph on your calculator determine the coordinates for each of the following points. All right? Over what interval is this function positive. All right, well, what I'd like you to do is play around with this, you know, put it into your calculator, graph it, look at the table, et cetera, and see if you can come up with the coordinates for all four of those points. Pause the video down. All right. Let's go through it. So we're going to need our calculator again. We'll make it the right size for the screen. And let's hit Y equals. Okay. I'm going to clear out the equation from the last problem. I'm going to enter this one. So we've got X squared. -11 X. Minus 26. All right. Once you get that equation in there, as always, look at it. Now we need subtraction signs, not negative signs. That would throw us off a lot, and wouldn't even give us an error this time. That would be bad. But it kind of looks good. Now I want to, I think I want to use the table. You know, who knows? I mean, if the points are integers, the table will work out well. I don't know exactly where to make it start, but let's pop into the table. Table setup, at least. Let's go into that. All right, now based just on the picture, it looks like I've got an X intercept that's negative and an X intercept that's positive. So let's, I don't know. Let's set our minimum to be negative 5. Okay? So it's not our minimum, but our table start value to be negative 5, and we'll go by ones. To begin with at least. So now, let's pop into our table, okay? Let's go into that. Then let's start scrolling down. We're looking for those zeros. And we see the first one pretty quickly. You know, see here when X is equal to negative two, Y is equal to zero. That's got to be point a so that's negative two, zero. That's our first one. All right. Let's find our other zero real quick. So we might have to look for a little while here. But not to hit the down button for a while. All right? Now, by the way, as we're doing this, you can kind of be looking at things like turning points and whatnot. I'm not going to record them yet. And there's my other zero. Okay? So my other zero, what I can see here, right on the table is 13 in the X column gives me zero in the Y column. So that's point B 13 comma zero. What we're really doing here is just exploring how we can use the graphing calculator to investigate the behavior of a quadratic function. All right, hey, let's find the Y intercept. That's really easy, right? The Y intercept is going to be the output when X is zero. So let's go up in the table. Get to X equals zero, easy enough. The Y intercept is zero, negative 26. And when a parabola is written in standard form, there's your Y intercept. All right, now let's take a look at the turning point. This is going to be the most challenging one. Let's kind of go down in the table. And here's why it's going to be challenging, right? Notice that at X equals 5, we have a Y value of negative 56. X equals 6, we have a Y value of negative 56. That means that the turning point has to lie immediately between them. 5.5. Now, we need to have the Y coordinate at 5.5 as well. So there's a lot of different ways we could do that. We could just evaluate the function of 5.5, but since we're kind of in that table frame of mind, let's use the table. I'm going to pop back into table setup right now. And I'm going to make my table start at 5. But I'm going to make it go by half unit increments. I know my turning points at 5.5. I just don't know what the Y coordinate is. So I'm going to make it go by halves. Got that? Now let's go back into the table. And they're right away. I can see the turning point, and I can tell it's at 5.5 negative 56.25. Okay. Last little piece, we don't really need our calculator for this. We have everything we need. It says at what interval is this function positive. At what interval is this function positive. All right. Well, positive just means it's going to be above the X axis. And it's above the X axis. Here. So that's X values over here. And it's above the X axis here. So X values out here. So it looks like it's positive whenever X is what? What was a? Less than negative two or when X is greater than 13. This is a great way to put it. I'm going to get rid of that little weirdness there because that almost looks like a negative. You can do it in interval notation. It would look like this negative infinity to negative two parentheses, not bracket. Then you use what's called the union symbol union, 13 to infinity. By the way, we can't include the 13 or the negative two. Here we don't have those equal signs. Because if those locations, Y is equal to zero. So it's not positive. It's neutral. All right. Pause the video now, write down anything you need to. All right, I'm going to get rid of the text. Now I think we can put our hard work and TI 84 plus away. And now let's finish up this lesson, all right? So in today's lesson, we really just reviewed some real basics about quadratic functions some terminology, like turning points, axis of symmetries, zeros, Y intercepts. We also looked at the shape of a parabola and what controlled whether it was open upwards or whether it opened downward. All of these are important as we move on. Now what we're going to be seguing to next is really, really, a look at most of the algebra associated with quadratic expressions. So make sure that you really kind of look at the homework on quadratic functions before further lessons where we look at their algebra. I'd like 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.
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