Calculus: Graphing Lines

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The Basics
Precalculus Review
Graphing Lines Page [1 of 6]
Well, so, now we know about functions, at least we know how to write them down and what not and I thought I would tell you a couple of things right now. The first thing is that there's a way of looking at functions, to actually visualize things and you know what they say, a picture is worth a thousand words. Okay, so anyway - so how can you look at a function and the answer is you can actually graph it. Let me just sort of quickly show you some of the basics of graphing.
You'll remember that when we graph things, we use the XY plane--looks something like this, and this is the X-axis and this is the Y-axis. And how do we plot points? Well, you know how we plot points. We go over this way in the positive X direction and this way is in the negative X direction; up is positive Y and down is negative Y. So for example if you want to plot the point let's say - five comma two. What you would do is five comma two, well; the first number is actually the X value. So you would go over five, so you'd go over one, two, three, four, five and then you'd go up two. One, two and then you would put a dot right there and that is five comma two. So it's always good to remember that we always write this as first the X and then the Y and you know that. Okay, so that's how you plot points.
Now how do you plot graphs? Well, what you would do is if you have an example of like, you know, F of X equals stuff. Well, what you do is you plot for each X value, five. You plug in to the machine and you see whatever it spits out and whatever it spits out, you put that in as the Y. Sometimes you might want to write Y equals F of X and you put up that height, whether it's up or down, depending if it's positive or negative, depending upon what the value is and then you get really pretty pictures. For example, you might get a really exotic picture, like that. Isn't that pretty? Look at that one. And that would be the graph of let's say, you know, Y equals F of X. You give me the X - look ______ you give me the X. This is what spits out and that spits out the Y value. You see how that works, you give me the X... So for example, I know that F of two - I'm sorry, F of five would have to be two. Right F of five would equal two. So that's a way of thinking about the pictures of graphs - and we're going to look a lot at this, by the way. We're going to look at so much; you're going to be so intimate with these pictures, by the way. Because there's one great question that I could ask you right now that you can start to chew on if you want. If I give you a function, how would you sketch the graph of it? That's an interesting question. What if I give you a really complicated function, how would you sketch the graph? Well, you know how we learned it when we were kids. You'd make a little table of X and Y and plug in and so forth, but that only gives you a few points. How do you get all the points? In fact, calculus is going to empower us to do that as well -- but I digress.
What I want to do instead of doing something really exotic is do something a little bit more concrete. I want to look at some very, very special examples of functions. I want to look at, in fact, the nicest kind of functions you can imagine. I want to look at lines - lines, the beauty and straightness of a line - very attractive, very soothing, very tantalizing, very tasty. For example, I show you the line. Look at that. A beautiful, delicious sesame breadstick. You can break it if you want and share it, which I never do. By the way, the breadsticks are real. These are very good, umm, and the little sesames are quite good. Anyway - I'll eat those later. Now that actually is - oh, look at the little sesames here. That actually is an example of a line, sort of a broken line, but I've got another one. You know I've got plenty of breadsticks. Do you want breadsticks? See me I'm getting them wholesale. They're actually thicker than a real line is.
A real line is - in fact, some of you in fancy restaurants might see these kind of breadsticks. This is still not capturing the spirit of the line because the line actually has no thickness. So even as delicious as these are - and these are really good and crunchy. Can you hear the crunch on that? Umm, it is delicious, but they capture the spirit of "lineness," until we draw them like this of course. But understand that actually a line has no thickness. A line just has basically a direction, which is straight. So, I'll put these culinary treats away for a second and clean up my dining room table here. Just come into my home, make yourself at home. Really, lines are objects that look like this. And you'll notice they have - well, an interesting form, because they're very, very straight, and yet there's something to them. There's about two things that I think are interesting about lines. See - what identifies that line out of all the universe of lines? How would we actually find that particular line?
Well, the answer is that I actually need two, two pieces of information in order to completely identify this line. And I'm going to tell you what those two pieces of information are right now, but you can probably figure it out. One is going to be pitch. For example, suppose I take a look at an axis here and I draw a line. Look at this. Ping. That is a beautiful line. But you see that line - I can make it look a little different by doing that. That's now a different line. The pitch is different. In fact, if you were climbing up this, this is going to be a mountain that's not so hard to climb but what about this one. Wow! Would you want to climb up that? I should say not. At least, I wouldn't because it would be really exhausting. This is very heavily pitched; whereas, this is less pitched. So one variable in figuring out a line is to know its pitch, which we call slope.
Now suppose we solidify the slope idea in our minds. Okay? Does that completely determine the line; does that completely determine my line here? I tell you it has that particular pitch; it has that particular slope. The answer is no. Can you see other lines that would have the same slope that yet would not be my line? Absolutely. There's that one. There's that one. I have a plethora of lines in fact. There's that one. There's even the famous green one and the black one. All these are lines, which have the exact same pitch but notice that they are not the same line. Why? Because they're shifted, right. One is here and this one is shifted up, it's shifted up. I admit they're all pitched the same. You wouldn't care which of these you had to climb because they all have sort of the same pitch to it, but they are all different because they are all positioned differently. So I need to have an anchor point. I need to have one point that I know is definitely on the line and that would anchor it. So if I said for example, "Well, it contains the point five two." Well, then I would go over five and up two and I would see, "Oh, okay, then it must be the blue one." And all the other ones, as good a guess as they were, in fact, would vanish, because now I know precisely which line is my line. Two pieces of information - the pitch and then a point.
Now there's a lot of ways of thinking about this. One way of thinking about this is to give me the pieces of information in the following way. To give me the pitch, which is slope and then to give me what's called the Y intercept. Let me remind you of the slope intercept form of lines - slope intercept. And in fact, you might even remember this from high school. And the way it works is - here's how you'd write it - Y equals MX plus B. Remember I told you that there are two, two pieces of information that are required for me to identify a line and there they are. M represents the slope, M for slopes, slopes, S, S, S, slope, that's why we call it M. Oh, okay. Don't ask me, I just teach the stuff, you know. Okay, slope. And this B represents where the line crosses the Y intercept; so, this is the Y intercept. So if you give me the Y intercept and you give me the slope, I can actually completely determine the line and write it in this form - Y equals MX plus B. These are numbers here and these are the variables.
So for example, let me show you how this would actually work in practice. In practice, the first thing you have to do is find M. So how do you find slope? Let me tell you what slope is. Slope, which we call M, is equal to - well, it captures the pitch, it captures the pitch. So, let me actually take one of these lines and show you how you would figure out what the slope of that line is. Well, to figure out the pitch, we're going to need two points, two points on the line. So let's say we have one point here. We'll call that - well, I'll call this X one because it's the first X and Y one, it's the first Y. So that's the point X one, Y one. And the second point, let me call this X two and I'll call this Y two. So that's two points and now you can see, there's only one line that connects them. It's that line right there; it's this nice blue line.
Now how do I find the slope? What I do is I look at the change in Ys and divide it by the change in X. So this equals the change in Ys - so that would be the difference in the Ys divided by the difference in the Xs. That's the definition of slope; so that's what slope means. That's exactly what slope means. Another way of writing this, by the way, is - well, I could write it, certainly fancier, but it's the exact same thing. Remember change we use the delta symbol, so in fact, I could write this as delta Y over delta X. The change in Y divided by the change in X. I sort of like this too, in fact. In fact, this actually leads me to a little question that I want to ask you and actually have you answer. Slope equals the change in Y over the change in X. Have you ever seen anything like this before? Is this slightly familiar to you? A little dejá vu. Well, take a second right now and see if this sounds familiar to you. We've seen it before together and in fact, this is an important connection with what we've already seen. So take a second right now and see if you can figure out the difference.
Well, remember this? Rate equals change in distance over change in time. Looks pretty much like slope except the M is played by R and tonight the Y role is played by D and the X role is played by T, but outside of that, it's the exact same thing. Is there a connection between rate and slope? Well, the answer is yes; it's an important connection. We're going to see a lot of it, but at least for now we see that it sort of makes sense because of how the line is steep is in some sense a rate. So in fact we're seeing already a connection between some things we've already talked about, really neat, really neat. But okay, so much of the neat stuff. That's how we find slope and now if you want to use the slope intercept form, if you want to use the slope intercept form for expressing a line, what you have to do, I remind you, is to find the slope and then figure out the Y intercept - where the line intercepts the Y axis.
So, let me actually do an example for you right now. Suppose that I know that the line passes through the point, let's say, three, two and minus one, four. What I'd like for us to do right now is to write that line in slope intercept form. Okay, so what do we have to do? Well, we have to find the slope and then we have to find the intercept.
Let me show you exactly how that goes. First of all to find the slope, what do we do? Well, the slope - change in Y over change in X. Okay, simple enough. So let's just take the changes in the Ys and the changes in the Xs and let's figure that out. So the slope would equal - well, the change in Y, so we have to subtract the Ys. Now let me tell you a little thing here that's really important. I don't care how you subtract. You could take the two and subtract the four. You could take the four and subtract the two. It's up to you. You know, free will, I believe in free will and this is free will. There's only one thing that I require of you and that is, that wherever you subtract first - you can subtract this Y from that Y, then you have to promise me that you're going to be consist and then subtract this X from that X. All that matters here is that if you take this and minus that then you take that and minus that. Don't swap in the middle of the game. You deal the hand and then we play. Okay. Now I'll cut the cards.
So if we look for the M, we take the change in Ys - so I'm going to take four and subtract the two. So that would be four minus two, not surprisingly and then I'll come back to here and take the minus one and subtract the three. So I have minus one minus three and I combine this four minus two is two minus, one minus three minus four. You cancel a little bit and see minus a half. So the slope of this line is minus a half.
What does it mean for a line to have negative slope? It means that you would love to walk through it because it's pitched down. If a slope is negative, the line is pitched down. If the slope is positive, the line is pitched up. Real quick question, I'm not going to ask you to answer it but see if you can do it right now on the fly. What happens if the slope is pitched zero? The slope is zero. How is the line pitched? Horizontally, cause it's walking on the street, flat, nice, smooth. If it's pitched - if a slope is positive, pitch is going up like this, in this direction; slope negative goes down. So this we can see is actually going down, going down.
And now what about the Y intercept, we have to find the Y intercept. Well, we have the slope and remember the formula for this line is going to be Y equals MX plus B. And our mission in life, by the way, in case you're wondering what our mission is, is to find those two pieces of information. We just found M; now we've got to find B. Well, how do you find the B? I'm not quite sure yet, but first let's celebrate the fact that we found the M. So let's plug that in right now before we forget, so we're getting close. We're half way there. We see Y equals minus one half. That's the slope we just found X plus B.
Now how do we find the B? What I do is I take either one of these points and I plug them in for X and Y because I know that those points are on the line. So since I know those points are on the line then if I plug in these two points; this for X and that for Y, or this for X and that for Y. I know this thing has to be true. It has to hold. And then I have all numbers except for the B and I can solve for it. You could pick either one. I'll use the minus one four, but actually it might be sort of fun if you're a little rusty on this stuff, to try this after I finish using the three, two and show you get the exact same value for B, no matter which. Sort of surprising, it doesn't make a difference what you plug in; you'll get the same answer for B. But I'll plug this in. So for X, I'm going to plug in minus one and for the Y, I'm going to plug in four. And this would give me four equals minus a half multiplied by minus one plus B. Well, minus a half times minus one is just a half and if I subtract the half to bring it to this side, that would four minus a half and so I would see that B equals four minus a half. So let's see, four I guess is eight halves. If I subtract one half, that leaves me with seven halves and so B equals seven halves. Might be sort of fun, like I said if you're rusty, try this exact same procedure with the three, two. Plug in three for X, two for Y and solve and you'll get the exact same value for B. And so our answer is that the line's equation is Y equals minus the half X plus seven over two. So there's the answer; there's the equation of the line. Do you have feeling for that? Do you feel that in your bones? No, you don't. Don't nod at me. It's equation. Well, who understands equations?
I like pictures. So, let me actually show you what that looks like. If you wanted to graph this thing, what would you do? Well, what I would do is I would start at my Y intercept, seven halves. So I have to go up seven halves which is what - that's three and half. So I go up three and half, so one, two, three, four - oops, went too much - three and a half right here. That's the Y intercept. That's the value that this line is going to cross the Y axis, but now remember, there are a lot of lines, there are a lot of lines folks, that go through that point. There are a lot of lines that have that as their Y intercept. Which one is our line? Well, we determine that by its slope, these are all pitched differently. So get rid of all those, we're going to find the one that's just right for us.
This one has slope minus a half, which means pitched down somehow. And then how do I find out what the pitch is? Well, if you remember, this is the change in Y over the change in X, so one way of speaking about this is to go minus one unit over in the Y direction, which would be down. So start here and go minus one unit down, would take me right to here and then that two tells me I have to go positive two in the X direction which would be if I mark this quickly, it's one, two. I go down one and go over one, two and that point must also be on my line and now, when someone asks you, "What's my line?" I know the answer. It's exactly that one. That is the graph of this line. That is the graph of the line that passes through three, two and minus one, four. In fact, the minus one, four, you can almost sort of see, look at my picture by the way. Minus one, see it. Look, it almost looks like it's going right at four, doesn't it? You see that? Minus one--let me see if I can sort of show it to you. See there's the four - oops, sorry, there's the four and it crosses right at minus one. My picture is actually amazingly accurate. The web is incredible. Okay, but I digress. Okay. So that answers this question and in three, two by the way, you can check that. One, two, three, two, one, two - look at it, perfect. Go up two, go over at three. I am a good graph drawer. So anyway, that's how we actually found the equation of the line that goes through those two points and we did it. Great. Now you'll notice one little thing about this. It was a lot of work; I'm sort of winded. Are you tired? Yada, yada, you have to find slope. You have to - then you have to find the B and so forth, right? But that's how we learned it. That's how we were taught what it is and that's right. But now let me show you a really easy way of writing equations for it.
Okay, so what's the easy way? Well, let me remind you of what we just did. We were looking at the slope intercept form. Great. Slope, Y intercept, you're home free, but what if the question were posed like this question was; namely, you just know two points. Well, how do we solve that? Well, first we found the slope. Okay, fine, and then we had to do all this gymnastics to find B. Well, there's actually a really easy way of doing this problem and it involves actually a different form of a line and in fact, I'm going to let you in on a little personal secret. This is my favorite way to express lines, and you know what, if you are staying with this program here - calculus - you are going to love this way. You are going to grow to love this way, just as much--if not more, than I do.
So what's that new way? Well, that new way is actually an old way. Don't you like it when you can recycle ideas? You know, you think of this sort of age of recycling as being sort of green and you're conserving and it's good, da da. But you know, in mathematics, we've been green for years. In fact, we're green with envy; no, that's just a joke. Okay, but the thing to look at here is this formula. You all know what slope is. It's this. Here it is again. This time I'm going to put it in a green frame. Okay, well, now instead of worrying about the Y intercept - what's so special about the Y intercept? What are you married to Y? Who cares? As long as we have an anchor point and as long as we know the slope, we completely know the line. So who cares if it's the Y intercept or not, just take any point on the line. If you know a point and you know slope, you should be done. You shouldn't have to go through all the hard work that you watched me go through just to get a lousy equation for a line. It's just not in vogue. So what do we do?
We come back to here and think to ourselves, "If we know slope and if we know one point, that gives us the line." So in particular, I'm going to tell you about my favorite way to express lines. Here it is, folks, you've been waiting for it. It's called Point Slope Form. Oh, I know, some of you really are going to like the Y intercept form and I'll tell you something. I was a believer in Y intercept form too, when I first learned it. So I'm like, nah, I don't want to do this, but you know what I've been doing this for years, folks. This is what you have got to grab onto. This is the way. Here it is. Y minus Y one equals M times X minus X one. That's it, that's the whole thing. That is the Y Intercept Form. That is the way to express a line. It's equivalent to the Slope Intercept Form but it's a slightly different thing. It looks a little different but it's equivalent. Let me tell you how this works. All you need to know is the slope. So this is the slope and X one Y one; just a point on the line, a point on the line. You give me a point on the line. You give me the slope and I plug into this formula. Notice the X and the Y - those are those variables - but this is a number, that's the number that represents the slope and this is just a point that's on there. So let me just show you the power of this, because you're saying - I can hear you - you're saying, "I sort of like the Slope Intercept Form. I'm not willing to switch yet."
Well, let's come back to that question that I first posed. Find an equation for the line that passes through these two points. Well, what do we need? Well, all we need is the slope and we need a point on the line. Well, the slope we have to actually find, just like we did before. So, let me remind you how we did that. We did that by taking the change in the Ys and the change in the Xs. So we did that already; I won't do that again. I'm making a mess of things here. Boy, if you were to see me here, you would not believe how messy it is. That's okay, I cleaned it up. I cleaned up. As long as you clean up your mess, I don't think it really matters. So I just found the slope by taking the change in Ys over the change in Xs, so I got a half. That wasn't so much. That's all we did there. That's all we did there. That was pretty easy. It was all this other stuff that was all complicated and yicky and everything here. But can you see me - thought I'd make sure. Hi, can you see me? Yeah, okay, okay, still here. Great.
So I just found the slope is minus a half, but instead of going through all this, what I'm going to do is jump right to the answer. Because the answer is - since I know the slope is minus a half and well, I need a point on the line. Well, I know two points on the line. I have twice as much information now as I need. I can pick either one of these. I'll pick this three, two, so I know three, two is on the line, is on the line and so I'm just going to write down the answer. No algebra. Why? Minus the Y value, which is two, equals the slope, which is minus the half times the quantity X minus that X value which is three. And that is an equation for the same line. Look at that. No work at all. I mean, don't you want to order this right now over the web? Get your credit card ready because this is what you want. Look at that. There was no algebra. I didn't solve for the B and do all those things and may be make a mistake and maybe I subtracted a side. I forget to subtract and so instead I add and na, na, na, na. No thinking here folks, that's why I like it. I don't like thinking and if you can actually stop me thinking and not do a little bit of arithmetic, I'm happy. Look at that. Had to find the slope, not a big deal, change in Y over change in X, just plugged in the point. Now maybe there's some of you that are still skeptics and saying, "You know, I like that other way of solving so much I'm not even convinced that these two answers are the same. They're supposed to represent the same line. Well, they don't look the same to me." All you've got to do is solve this for Y.
If you take that negative two and bring it over, let's see what you get. I'm going to do this really fast, just for you. Just so that in case you're a skeptic. In case you right there, are a skeptic, let me just show you that I'm not a complete wacko. So Y equals - I'm going to bring this two over so I have to add it to this side. So I have to add two, I'm also going to distribute that minus a half everywhere through there. I hope you don't mind that. So I have minus a half X and then I have a minus a half times minus three. That gives me actually a net gain of plus three halves. Then I'm going to bring that two over so I have to add the two. That's an extra two I get to add in there; that's a lot I'm adding. What do I get? I get minus a half X and then I take two and add it to three halves. Well, two is actually four halves. Four halves and three halves, seven halves - you've heard that line before. You sure have because that answer you can see, is identical to that really, really arduous way of getting the answer.
So here's a really, really neat way of getting a different but equivalent form of a line. All you need to know is two pieces of information - you need to know slope and you need to know a point on the line. If you've got that, you just use point slope form. Put in the slope here, drop in the point right there and you are home free. You are sitting pretty. Now that's lines. What about more exotic things? What about curves? Well, we'll take a look at that next, but right now all you have to solve is lines. See you in a bit.