This unit is all about understanding linear functions and using them to model real world scenarios. Fluency in interpreting the parameters of linear functions is emphasized as well as setting up linear functions to model a variety of situations. Linear inequalities are also taught. The unit ends with a introduction to sequences with an emphasis on arithmetic.
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 four lesson four, more work graphing lines. Before we begin, let me remind you that you can get a copy of the worksheet used in this lesson, along with a homework set by clicking on the video's description, or by visiting our website at WWW dot E math instruction dot com. Don't forget about the QR codes on our worksheets, scan those with your smartphone or your tablet to be taken right to this video. All right, let's begin. In the last lesson, we looked at graphing non proportional linear relationships. In other words, graphing lines that don't go through the origin. Today, we're just going to work more with those ideas specifically with slope and Y intercept and make sure that we really happen down. I like this first problem a lot, because what it's given us is a graph with no scales on it whatsoever, and four lines drawn. Then I give you four equations, and I want you to figure out which one goes with each one of the graphs. So what I'd like you to do right now is pause the video, take maybe 5 minutes, use your knowledge about slope and Y intercept and try to figure out which one of those lines goes with each one of those four equations. Pause the video now, please. All right, let's go through them. So here's the key. Let's do a little review. All right. When we have lines that are in the form of Y equals MX plus B then the M right? The M is the slope. And the B is the Y intercept, where it crosses the Y axis. All right. So let's take a look at line one, if you will. Line one. This guy. What do we know about it? Well, one thing that I know immediately is that it's Y intercept. Is a positive number. All right? That means it can't be this one and it can't be this one because these two have negative Y intercepts. As well, I know that the line is always increasing. Right? It's always getting larger. The Y values. And therefore, I know that the slope is also positive, which means it can't be this one. So line one, by default, has to be this line. Now, just to make sure, write this line has a slope of one, even though it may not be written and a Y intercept of 5. So it's got a positive slope and a positive Y intercept. Now, if you need to, pause the video again right now to see if you can get the other three. All right, let's go through them. So line two, what do we know about line two? What we know about line two, let me do it in a different color. Oops, that's not what I wanted. Let me do it in a different color. Line two has got a negative Y intercept, but it continues to have a positive slope. So line two, the B is negative, but the M is positive. Which one has that hot? It's this one, right? Here's a positive slope. And a negative Y intercept. So that's got to be line two. On the other hand, let's go with line three. Line three, what do we know? Well, it's got a nice positive Y intercept. Because it's above the X axis. But unlike the other two lines, this line is decreasing. It's forever decreasing as X increases. And therefore, the slope must be negative. So that's got to be this choice, right? A positive Y intercept, but a negative slope. Now obviously, by process of elimination, we know which one is choice four, but let's go with it anyway. So line four what we have is we've got a Y intercept that is negative and a slope, well, here's a line that also decreases. So we've got a slope that's also negative. So this must be choice four, right? Because we've got a negative Y intercept and a negative slope. So it's kind of cool. And this is important because in the future, if you see a line, let's say that you had a line like Y equals negative 5 X negative 5 X plus 7. And you needed to make a quick sketch of it. Well, you would know that it's Y intercept was positive. And you'd know that it's slope was negative, and therefore that would be a reasonable just first sketch of the line because it's one that has a positive Y intercept and a negative slope. All right? So these are very, very important parameters of linear functions. The slope and the Y intercept and there are things that you should be able to grasp pretty quickly. All right, I'm going to pause the video now, or you can pause the video now because I'm going to clear out the text in just a second, okay? All right, here it goes. Let's keep going. Remember, the whole focus of today is to just become comfortable with linear functions and Y equals MX plus B four, right? And understand those two linear parameters. The slope and the Y intercept. Exercise two. On the grid below, graph the equation Y equals three halves X minus three. First, identified slope and Y intercept. The graph. All right. Now, again, because this is review from both what we've done in this course, and from previous courses. What I'd like you to do is pause the video now. Identify the slope and the Y intercept of this line, and try to plot it. If you're struggling with that at all, don't worry, obviously we're going to go ahead and show you how to do it. But try to see if you remember how to do it first. All right, let's go through. Since this thing is already written in the convenient form, Y equals MX plus B then it's pretty easy to identify. I'm going to go back to blue just so there's a little bit more contrast. It's pretty easy to identify the fact that the slope is three halves. And that the Y intercept is negative three. All right, so the slope is three halves and the Y intercept is negative three. Now the way that we use those to plot is, well, we first plot that starting point at negative three. And then we use the slope. Now remember the slope is just the average rate of change. It's the change in Y divided by the change in X, which means that we're going to go up three every time we go to the right two. So we go to the right two, and then we go up one, two, three. And then we go to the right two and go up one, two, three. To the right two, one, two, three. Now real quick, by the way, just a side. Right? As you all know, a negative divided by a negative is the positive. So in a certain sense, we could look at the slope as being negative three divided by negative two. That's the same thing as three divided by two. But that would tell us that we would go down three and to the left two. So one, two, three, one, two, and there's our line. Let me use my line tool right now to graph it. Let's see if I can get a good graph for once. And not bad. Extend it maybe a little bit up this way and then some arrows. All right, there's our line. Many, many different functions exist in mathematics. Most of which in order to graph, you have to construct a table of values, sometimes a very intricate table of values, but not lines. Lines, we all hope you can rearrange into Y equals MX plus B form. Identify the slope, identify the Y intercept, and plot them pretty accurately. Let's take a look at number three. Number three says to write down two points on this two points this line passes through and use them to calculate the average rate of change of this function. Just so that we're all in the same page, let's choose the same two points. Why don't we grab this one? And why don't we grab this one? This one has coordinates one, two, three, four, one, two, three. So that's at four three. Let me write it down here. And this one has coordinates one, two, one, two, three, four, 5, 6, so negative two, negative 6. As it's coordinates. Now remember the average rate of change is F of B minus F of a divided by B minus a it really doesn't really matter which one we consider a and B, right? This is, again, exactly the same as the slope formula, Y two minus Y one. Divided by X two minus X one. I like to keep this point as being my first one and this one is being my second one, even though I wrote them down a little bit differently. Because the way that I look at rate of change is this idea of actually traveling along the line like this, right? From the point negative two negative 6 to the .43. So if we do that, remember what we're doing is we're looking at the change in Y, right? Three minus negative 6 divided by the change in X for minus negative two. Three minus negative 6 is positive 9, right? Because a negative and a negative make a positive. And same deal here, 6. Now I want you to pause just for a minute though because I do want you to understand this. This isn't just a formula, right? I mean, it is a formula. But really kind of understand what's going on here. We'll stay in red. What we're really saying is to go from this point to this point, we have to increase our X by 6. And we have to increase our Y by 9, right? It's really just a little right triangle. Now, if we reduce this, though, and divide both numerator and denominator by three, we get three halves. Which of course we must get because that's the slope of the line. And that was really the point of exercise three. What makes linear functions linear functions is that they have a constant average rate of change. What makes other functions not linear is that they don't. So for instance, we'll be looking at parabolas a lot later on in the course. Their rate of change depends on what two values or what two points you pick on the function. Lines it just doesn't matter. Their average rate of change will always be that constant slope. All right, I'm going to clear out the screen, so pause it now if you need to. All right, here we go. Let's move on. Now, the very often we are given linear equations, equations that when graph look like lines, and again, those are very easy to identify linear equations are ones where X and Y are simply to the first power. Nothing higher than the first power. So for instance, let's take a look at exercise four before we even jump into it. This is a linear equation because both the X and the Y variables are to the first first power. But it is not in Y equals MX plus B four, which is kind of a pain. And yet we need it to be and Y equals MX plus B form if we're going to identify the slope in the Y intercept. So the first part of this problem is just rearranging this thing into Y equals MX plus B form by using different properties. And we at the common core level need to be able to justify what we're doing in math. So we've done this before. What I'd like you to do is try to justify each step in this rearrangement either with a property of equality. Or a property of numbers. Okay? Pause the video now and see if you can do this. All right, let's go through it. So we took this equation. First thing that we did was we added something to both sides. That's called the additive. Property. Of equality. The additive property of equal of equality. The second thing that we did is we took this 12 plus 6 X and we made it into 6 X plus 12. That is the commutative property. The commutative property. Technically, it's the commutative property of addition. So I'm going to write that down. Because there's two commutative properties. There's one for addition, and there's one for multiplication. The next step is that we took and divided both sides of the equation by two. This is what's known as the multiplicative property. I didn't quite get that first eye in there. I think we're just going to go with prop. Of equality. Now remember, if you're confused by that, you might say to yourself, wait a second, we didn't multiply we divided. But what we really did is we multiplied by one half on both sides. So that works also with subtraction and the additive property. All right? Then we took that division by two, and we divided both the 6 X by two, and the 12 by two, and that. Is the distributive property. All right. And finally, we got three X plus 6, right? So we're able to take an equation like this and rewrite it like this. Now remember when you do that, then what's happening is that you're getting equivalent. Equations. Equivalent equations are equations that have the same solution set. So whatever solutions there are to two Y -6 X equals 12 are the same solutions to Y equals three X plus 6. And of course, letter B is pretty easy, identify the slope and the Y intercept of the line, pause the video real quick and do that for me. All right. So hopefully not a problem. M the slope. Is equal to three, B, the Y intercept is equal to 6. All right. So very often, we're given linear equations in the form that we see up here. All right? Or more appropriately, we're simply given linear equations and forms that aren't Y equals MX plus B but since we want to be able to talk about lines, graph lines, think about whether they're increasing or decreasing where they start, I either Y intercept, we very often want to rearrange them into Y equals MX plus B four. All right. We're getting a lot of practice with that in the next exercise. So pause the video now. All right, I'm going to clear it out. Okay. Let's do some practice rearranging equations. All right, exercise 5, rearrange each of the following linear equations into Y pulse MX plus B form and identify the slope and the Y intercept. Okay. Let's do the first one together. And then I'm going to let you do then the next three on your own. But throughout it all, what I'm going to be doing is I'm going to be continually emphasizing these properties of equality. So, for instance, in letter a, the first thing I would do is I would add three X to both sides. All right, and of course negative three X positive three X cancel and we get 15 plus three X on the side. All right, so that's the additive property of equality. Then I use a little commutative property of addition and swing this around this way. The reason that I want to do that is I want to get it into Y equals MX plus B form, so I'd like the MX plus B or it to be MX plus B and not B plus MX. I have to get why all by itself, so I'm going to use the multiplicative property of equality. And multiply each side by one third, if you will. And then I think what I'm going to do is immediately use the distributive property and divide both of those quantities by three, of course, three divided by three is one. And 15 divided by three is 5. So here's my first step. Now be careful, write the slope just simply one. The Y intercept is 5. Remember, slope is a number. It's an average rate of change. One of the largest mistakes I'll see students make, and it's a very simple one, is they'll include the X so you'll say something like, what's the slope and a student will say, two X no, the slope is two, right? Here it's weird because there doesn't even appear to be a number multiplying X you are always welcome to put a one in a multiplication just as you're always welcome to put plus zero anywhere. But doing one times X shows us that we've got a slope of one. All right. How about this? How about pause the video now? Try letter be on your own, and then maybe we'll do letter C together and then you can try letter D on your own. All right, let's go through letter B so let it be very, very similar. Although the numbers are a little bit uglier, more negatives involved, we get two Y is negative 8 -5 X, then we'll use some commutative property to rewrite that as negative 5 X -8. Then we'll use the multiplicative property of equality. Now, I'm going to also do that distribution right away. This will confuse some students because they'll say to themselves well, two doesn't go into 5 very nicely. And I agree. But that's why we'll just leave it as negative 5 halves, right? We'll do negative 5 divided by two. That's actually the associative property of multiplication. So we're going to sort of choose to do that first, and then multiply by X second. Here it's rather nice because 8 divided by two is four. So there's our equation of the line and Y equals MX plus B four, and the slope is negative 5 halves. And the Y intercept is negative four. All right. Let's take a look at letter C now, letter C and letter D are a bit different than a and B only in one sense. If you look back at letter a, let's say, and you really examine, let me do it in red. And you examine this form. I know it sounds weird, but notice how the Y comes first. Same thing happened here. The Y came first. In these two examples, the X is setting here first, and that's sometimes all it takes to make students confused. All it takes. So for me personally, and this is me, the first thing that I would do in this problem is I would rewrite it as negative three Y whoops. I got a fly in here. Negative three Y plus X is equal to 6. All right. So in other words, I'm going to use the commutative property commutative property to flip that around. That just helps me because it makes it look a lot more like a and B then I can start to do exactly what I did before. Use the additive property of equality to subtract X from both sides or add a negative X depending on how you want to look at it. I can then use the commutative property again to rewrite this as negative X plus 6, all right. Little multiplicative property of equality allows me to divide by negative three on both sides. And now distributing. All right, this is going to be a little bit confusing, but we'll get it. All right, we distribute that division by negative three. Now what do we do with this? Well, again, I can put a one here. That's not a problem. And then what I'm going to do is I'm going to look at this negative one divided by negative three, and I'm going to make that into a positive. Remember a negative divided by a negative is a positive, a positive one third. Now here, sorry, that was a positive right there. 6 divided by negative three is negative two, so it's just going to be one third X minus two. Little mixed color there. I'm sure isn't too problematic, but that then tells me that my slope is one third. And my Y intercept. Is negative two. All right. Letter D is going to be very, very similar to letter C so what I'd like you to do now is pause the video, try to do that one on your own, try to simplify everything you can simplify. And then we'll go through it. All right, let's do it. So first things first, I'm going to rewrite that left hand side by using the commutative property. I'm going to rewrite it as I don't know how that just happened. A few too many few too many wires. So negative four Y plus 6 X is negative 20. All right, we'll subtract 6 X from both sides. Gives me negative four Y equals negative 20 -6 X while then use the commutative property to rewrite that as a negative 6 X -20. Multiplicative property allows me to divide both sides by negative four. Now let's do some distribution. Let's do negative 6 divided by negative four times X plus negative 20 divided by negative four. I think I'll complete this up here. Negative divided by negatives of positive, right? So we get 6 fourths X we're going to reduce that in a second. Negative divided by negative is positive. That one's easy. 20 divided by four is 5. And then of course we can just do a little bit of middle school fraction, reducing here. And get Y equals three halves X plus 5. There's our MX plus B form. With our slope being three halves, and our Y intercept being 5. All right. That almost looks like a three. That's a little better. Okay. There's a lot of writing on the screen. I assume right now my head is really tiny and probably somewhere right about up in here. Hi. Anyway, so write down what you need to at this point. Okay, pause the video. This is an important screen, and then we're going to finish up the lesson. Okay, let me clear out the text. So today's lesson, we were just really trying to reinforce what we should have already known about linear functions, and especially about the two important parameters of linear functions, the slope, and the Y intercept. We also saw today how to rearrange equations that are linear to get them into Y equals MX plus B form. This is going to be an important skill that's going to rise here or there in the course. So make sure that you get some work on it. For now though, let me 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. Remove Ads |
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