Graphing Review - Video Tutorials & Practice Problems

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1

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Plotting Points on a Graph

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Hey guys, I know a lot of you haven't seen graphs for a while. So I'm including this review to refresh you on a lot of those concepts and if you feel a little more comfortable with the math, I still suggest watching it. You might get some value out of it. The first thing we've got here is our two variable graph on the left. Okay. We're going to learn what the key parts of the graph are and how to plot points on the graph in this video. So first we're gonna label what are called the axis of the graph. So here on the bottom, what I've just highlighted in red, this is called the X axis for our X values and on the other axis going up we have the y axis. Cool. Generally we're gonna have numbers or values that tell us how much each jump in the graph is and for now we're just gonna go 1234567. Right? And usually when I give you graphs in this class, um you're going to not have to do this right? I'll have done it already. I just figured the first time we do it together, just so you get a feel for it. So here on the right, I have what's labeled a demand schedule and it's got some prices and some quantities, right? So at certain prices there's gonna be certain quantities that are demanded and later in the course we're gonna dive into these topics more. But for now what I wanna do is get these points onto the graph. Right? So the first thing we have to do is we're gonna label one of our axes as the as the price and one of our axes is the quantity in economics, we tend to label the Y axis as price and the X. Axis as quantity. This is how they've been doing it. This is just the convention in economics that's been around for a long time. So this is how we will do it as well, Price on the vertical axis and quantity on the horizontal axis. So let's go ahead and get these pairs what we call pairs of numbers and we're gonna plot them on the graph. So let's start with the first one which I'll call, let's label them all A B C D. E. Just so we know which one we're talking about when we're on the graph. So let's start with 61 a price of six and one demanded. So I'm gonna go to my price axis and find six right up here and I'm gonna start going out, right and then when I go to the quantity I'm gonna find one And I'm gonna go up from there and I'm gonna find the point where these two cross with each other. Right? So right here that's gonna end up being the .6 for price and one for quantity right there. I'm gonna erase some of those extra. Cool. So that's gonna be point a right there. Let's go ahead, I'm not gonna change colors because I'm I'm not gonna have five different colors for this right now I don't think we'll need it but let's go ahead and plot the rest of these. So actually for this one I'll use blue and then I'll rotate back to read just just to keep it a little consistent. So um here we have a price of five and a quantity of two. So we'll find five on our price access to on our quantity axis and we'll find the place where they mix or where they meet and that's gonna be right there. Point B. Let's go back to red and we'll do the same thing for C. So now we've got a price of four and a quantity of three right and that'll be point. See right there. Now point D. I'm gonna do in blue and that's got a price of three and a quantity of four. So you can see these are kind of the places where you can get mixed up which way you know which axes do I put the three, which one do I put the four? So you just have to make sure that you're on the right access when you start counting. So that'll be point D right there and let's finish it up with E. At a price of two and a quantity of five right there. That's point B. Right there. Cool. So that's how we plot stuff onto the graph I guess I'll put this as a blue B. And a blue D. Just to match what we've got going there. Cool. Alright, let's move on to the next example.

2

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Drawing and Shifting Curves

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7m

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So now we're gonna learn how to take points on the graph and turn them into a curve as well as how to shift that curve on the graph. We're also gonna learn how to shift curves just visually with no math, the tool use quite often in this class. So let's go ahead and look on the graph here. You'll see I have the points from the previous video where we learned how to put points on the graph. I've got those points here already. So when we're turning points into a curve, um what we do is we start at the left most point and we work our way right words. Okay. Um This one seems pretty simple, it's just gonna make a line and yes, a line is a curve, it's just a straight curve. So this is what this curve may look like right here in green. Alright. Um I just want to make an example here. I'm gonna do something right here on top. Let's say we had points that looked like this, ignore the other points right now. But let's say there were points like this, there's a specific way we want to connect those right? We want to start left to right, just like I said um you never want to double back and start going back to the left there back to the right. Um Let me show you in an example here, right? So these points we would connect them something like this right? And I don't want you to get confused and connect them. Maybe like this right? That's not how we would connect these points. Um you start at the left most point and you go to the right. Cool. So now let's talk about shifting this green curve right here. How do we shift it on the graph? Let's say someone told us we had to shift this curve, let's say two units to the right, two units to the right. Okay. So how do we do that, what we're gonna do? And the easiest way I find to do it is I pick the left most point. So in this case it would be this point that I'm gonna circle here in black right and we're gonna move it two spaces to the right, so I'm gonna count here. Two, that's one and that's two right there. That's gonna be our new point that I'll put in blue. Cool. So you do that with your left most point and I'd like to just go straight to the right most point and do the same thing, grab my pen And I'll pick this right most point right down here and I'm gonna move it two spaces to the right one to write and I'm gonna put my point right there, my new point and now that I have two points. Um if you just connect these two points you'll have your new line so I'll do that one in green as well. So here we go connecting these two points, we've got our shifted curve. So this new green curve right here, it's shifted to the right, Actually, I'm gonna do it in blue so we can see which one's which. So blue curve has been shifted to to the right. Cool. And a lot of times in this class, like I said, we're gonna be doing shifting of curves just visually, we're not going to put any math behind it, we're gonna have a reason, we're shifting the curve and then we're gonna have to see what happens after we've shifted the curve. Um And when I say, see what happens, we're gonna see what happened to the new price and the new quantity. Um but we'll get more into that in the next chapter. So I'm gonna draw a couple of graphs here just to explain what I mean by shifting visually. So a lot of times on a test or on a practice problem, you're just gonna kind of draw a graph kind of willy nilly like this, and a lot of them are gonna be graphs that look like this, we're gonna have an X. And remember I suggested having at least two colors. Um and we're gonna use those quite often. So in this case, um what we're gonna do is kind of like we did above on the graph, we're gonna shift the red line to the right, so now it's not two units, we're just shifting to the right, cool. So what you do is you start and you're going to pick a point on the graph, you're gonna move it to the right and then you're going to draw a parallel line just like that. Right? So when we do these kinds of shifts, what we're doing is looking for these points of intersection where this was the point of intersection originally. Now we're at this point of intersection here. Right? So we would be judging what happened to the price and what happened to the quantity after this shift. Right. So we can make assessments of that just visually without doing any math. Um but we'll deal more with analyzing it when the time comes. Now, I'm just trying to expose you to shifting the graphs like that. So let's do a couple more examples here. Now let's shift the red line to the left. So we're gonna start with the red line and you can see now if I went to the left, I kind of end up off the graph here. Right? So maybe I can pick a point like a little further down and go to the left here and now I can draw a parallel line. Right? So you just want to make sure you're going the correct direction and let's see where our new intersection is. We were here before and now we've moved down there. Right? So we will be able to make assessments about price and quantity based on that movement. Cool. A couple more examples here. So now let's move the blue line. Let's see what happens when we move the blue line to the right. So same thing, we're gonna pick a point here, move it directly to the right and draw a parallel line. Cool. So the point of intersection was there and now it's there. All right. Now let me get out of the way. I'm gonna do one more example here in this last corner. So sometimes we actually have to shift both the lines on the same graph and that's when it starts to get a little difficult remembering which line was which. So I like to draw my ex so that the original point of intersection is right in the middle, right? When I'm doing it visually I just keep that point of intersection in the middle. And then I'm gonna look at my new points of intersection. So let's say we had to move um let's just move them both to the right in this example. So first let's move the red line to the right. So I'm gonna pick a point here, move it to the right and draw my new line. And now I'm gonna draw the blue line. We're also gonna shift to the right. So pick a point, go to the right and draw my new line. So you can see now there's a lot of points of intersection here, it's like which one is the new one, which one is the old one. So you gotta be careful and pick the correct intersection here. So you want to have good eyes at finding these because this is gonna happen quite a bit in this class when we're studying demand and supply. So which one did you pick? This is gonna be our new point of intersection right here? Let me go. Yeah, I'll leave it in green. So we see that that's the new point of intersection right there. Cool. Um So this is how we're gonna shift curves visually on the graph and we can also do it mathematically like we did on the left. But mostly we're gonna do it like we did on the right in this course. See you in the next video.

3

concept

Calculating Slope of a Straight Line

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7m

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All right, so let's continue by learning how to calculate the slope of a line. You guys might remember this from algebra. Right. Um So here in the green box, I've got our our formula for slope, it's going to be our rise over run. Um The change in Y over the change in X Y two minus Y one over X two minus x one. These all mean the same thing in this class. It's probably not gonna get so algebra heavy. So we're just going to stick with rise over run for now. Cool. So let's go ahead and calculate a few examples here and see how this formula works. So let's start with this first graph a with the red line. Um And my first step when I'm calculating slope on a graph is I try and find two points that intersect on the graph, directly on one of these intersections on the graph. This first graph actually has quite a few of them. Um So, you know, right here, right here, right here, right. Some of our other examples won't be so easy to find those points, but the idea is we're gonna pick two of those points and we're gonna calculate our slope. So I'm gonna go ahead and pick this point and this 20.0.2 points uh that intersect the line there and let's go ahead and calculate the rise first. And then we'll do the run. So to calculate the rise, we have to see what the change in the vertical axis is, What is the change in the y value? So if we start here, we want to see what, how much did the up and down change for between these two points. So we started here at five and it looks like the next point is down here on four. Right? So it looks like we went down one and when we're calculating slope down is going to be negative and up is positive. Um just like when we're going left and right, left is negative and right is positive. I'll write that all down here, I'll put it here on the left hand side for you. So up is going to be positive and right is going to be positive, down is going to be negative and left is going to be negative up into the right is positive. Cool. Um So in our example here, like I said, we went down one, so that is going to be a slope of negative one. Excuse me not a slope of negative one. A rise of negative one. Now let's see what the run is. So from one point to the next, the x value seems to have shifted 1, 2 right here. So when it goes to the right, it's positive, right? We've got a positive one for the run. So let's go ahead and calculate the slope here, we've got slope and I'll write it rise over run. So our rise in this case was negative one. Our run was one. So that's going to simplify to negative one. Our slope here is negative one. Um, if you guys have a little, need a little refresher with fractions as well. I'm also including a fractions review uh, in this section too cool. So let's move on to part B here and let's calculate the slope here. I'm gonna get out of the way so we can see the example. And let's go ahead. Remember, like I said, the first step, we want to find two points that are intersecting the graph uh, at one of those intersections, right? So you can see in this, in this case we've got a few points here that don't exactly cross at those intersections. We want to find the two points or any two points that are crossing. It just makes it easier to calculate. So right here in the middle we've got one point and I'm gonna pick this one right here on the end and we're going to calculate the slope between those two points. So let's first do the rise the rise in this case. So it looks like we started at a vertical value of two, and the next point is at a vertical value of three. So let's see, we're gonna draw our arrow here and it looks like it went up one from 2-3. So I will write one right here and now let's do our run. So we started at three and we went to six. So it looks like our change was three here, right from 3 to 6 are run is three. So let's go ahead and calculate the slope slope again. I'll write it here rise over run. And in this case our rise was one are run was three and that's it. The answer is one third. The slope of this line is one third. So let's scroll down here. Um We've got one more graph, part C. And let's go ahead and calculate this slope. So I guess I'll come back so you don't feel so lonely? Hey guys, alright, so let's do part C again. We want to find two points where it's intersecting directly uh there on the graph. So notice kind of a point like that. They're not so easy to calculate. So let's find the easy points. I'll do in blue. We've got one right here and one right here, there's other ones. But those are the ones I'm gonna use. Cool. So let's start with our rise again. In this case we start at four, our next vertical value is six. So it looks like we're gonna go up here and it looks like we went up to right, we started at four, went to six. So our rise was to do the same thing with our run in this case it looks like we started at three and we got to four. So it looks like our run is going to be one in this case. And let's calculate the slope. So our slope again rise over run right. And our rise was to r run was one to over one. That's just too. So our slope in this case is too So let's go ahead and compare uh just let's look at these lines and see the difference in the slope and what the line looks like. So in part a we've got a negative slope, right, Our slope was negative one and notice how this line looks compared to the other lines, right? It looks like when we go from left to right, it looks like this uh this line is going downhill right, because the slope is negative, it forms a downhill right going from left to right and notice our other two which have positive slopes, they look like they're going uphill right, B and C both have this uphill tendency. But now let's look at one more thing here, notice and be our slope was one third and see our slope is too right. So too is quite a bit bigger than one third. And look at how these lines look right and be you kind of see like a soft growth here, right? It's kind of a little bit of an uphill where in see where we have a slope of two. It's a lot steeper. So the higher the slope is the steeper it's going to get this way and if it was a really negative number. So if a was negative to you could imagine it would be a lot steeper going down. Cool, Alright, so that's how we calculate slope. Let's move on.

4

concept

Calculating Slope of a Curve:Point Method

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3m

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Alright. So now we're gonna learn how to calculate the slope of a curve when it's not a straight line. So the idea here on this, on this graph, what you see is a curve um that's not straight, but in this situation, how do we calculate the slope? So when you see a curve like this, the slope is actually changing all the time. Um So you're gonna have a different slope at this point where it's rising pretty fast then at this point where it's kinda going up a little slower, right, you would imagine from our last video that those would have two different slopes. So the first method when calculating the slope of of a curve is to use what's called the point method. And what we do is we draw a tangent line right? Um uh We draw a tangent line. So a tangent line touches the curve at only one point. Okay, so the idea is we're going to calculate the slope of the line at that point on the graph. Cool. So once we draw the tangent line we just calculate the slope of the tangent line and then we know what the slope is at that point. So I'm gonna go ahead and do my best to draw a tangent line. It's not very easy to do this by hand. Um If you were ever to have to calculate this in this class, I'm sure they would give you the tangent line already. Um And I'll do my best here should look something like that. So the idea is that it's only touching the graph at one point even if it doesn't look like it from my example I did my best but the idea is that it's only touching the graph right there. So the tangent line is just going going going, it touches the graph and it keeps going just one one point that it touches the graph. So now that we have a tangent line we can go ahead and calculate the slope of the tangent line and we will know the slope at that point. So um using our same method from finding the slope, let's find to points that intersect the graph. Um And it'll make it easier to calculate. So I see one there um here's another one right here let's go ahead and calculate that slope. So um it looks like from the first point to the second point we are going up right let me do this in a different color. We'll do it in green. Um It looks like we're going up and from that point to that point we went up from 4 to 6. So it looks like our rise was two. And let's do our run now. So it looks like we started with an X. Value of three. We got to an X. Value of five. So our run was also too. So let's calculate this slope. So the slope of the tangent line. I'll put slope of tangent line equals still that rise over the run. And in this case we've got a rise of two, a run of two and that simplifies to over two simplifies to one. So the slope of the tangent line is one. And that means that the slope at this point where we drew the tangent line right here, the slope of that point equals one. So the slope of that curve at that point is one. Remember it's constantly changing, but at that point the slope is one. So that is how we calculate the slope of a curve. Using the point method, let's move on to the next video.

5

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Calculating Slope of a Curve:Arc Method

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3m

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Alright, so now we're going to calculate the slope of a curve that's not a straight line using the arc method. So, what I mean by the arc method is basically, we're gonna find this slope between two points, right? When we did the point method, it was just one point. Right? So now we can find the slope over a region like this. Okay, So we got instructions here on the right, we're going to draw a line connecting the ends of the arc. Right? So the region that you want to to calculate the slope over? Um Right, I've got two points on the graph there. I could easily calculate the slope over, you know, this whole region here, or this region like that, or this region here. Right? You're gonna get different answers in all those cases, because the slope is constantly changing. Um And you can pick any points, right? I just picked points where we've got intersection. It's just gonna make the math easier. So, here, this is the slope, this slope is the average average slope, average slope over the region over that ark. So, between those two points that you select and calculate, you're gonna be calculating the average slope over that region, not just what is the slope? Remember that slope is constantly changing. So, we're doing our best, we're gonna find an average. So, what I'm gonna do is I'm gonna pick these two points right here. Um And I'm going to draw a line connecting those points. So from here to here, we'll draw a straight line. So you can see that that almost approximates what what the graph is actually doing, right? So, this is why we're kind of finding an average, we're doing our best uh to estimate what that slope is. So, it's almost like we've got a line here going like this, right? We've calculated those points and we're gonna have this line going like that. So, let's go ahead and calculate the slope of that line. So, same thing. We're gonna do our rise and our run. So, let's see what our rise was between these two points looks like it goes up. We started at uh excuse me at four and we went up to five. So it looks like our rise was one. And let's do the same thing for our run. So we started at three, it looks like we went over to six. So it looks like a run was three. Right? So using our same formula for slope. So the slope of that line um connecting that arc slope is going to equal our rise over our run. Our rise was one are run is three, our slope is one third. So that is the average slope over that arc, right? And if we picked two other points, we would have got a different answer. But the method stays the same. You draw a line, you calculate the slope of that line and that will be the average slope over that section. This is the method that we'll use more often in this class. I'm not expecting you to have to be drawing tangent lines and stuff like that. So just be pretty comfortable with this. Being able to pick two points, draw the line and calculate the slope. Cool. Let's move on.

6

concept

Finding Maximum and Minimum on a Graph

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2m

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Alright, so now we're gonna do a quick recap on how to find the maximum point on a graph and how to find the minimum point on a graph. So here on this example of, I've got this kind of upside down you. Um So how do we find the maximum or minimum point? So what you'll see is that not all graphs have a maximum or a minimum? Um It's only when they kind of turn around like this, right where they're going up up, up, up up and then they turn around and start going down, right, so when we want to find the maximum point, it's that point where it turns around. So if you notice here the graph seems to be rising, rising, rising, rising, rising, rising, rising. And then on this side it's falling. Falling, falling. Right, so we gotta find that point where it turns around, so notice here it's still rising a little bit, right, it's still rising a little bit and then here it's pretty clear is the point where it turns around, I'm gonna do it in a different color there. So right here is the point where it turns around we're not doing any math here, I just want to be able to identify the maximum and the minimum. So right here that is our maximum. Okay, you're gonna wanna be able to do this um and find maximums and minimums on a graph. So what you'll notice is this one doesn't have a minimum. Um It went up up up to a point and then started going down down, down there wasn't a point where it was adam max bottom or max top, you might think this is a minimum here, this is a minimum here. Um but usually these graphs are gonna continue, so it would continue going down and it would be, you know, there wouldn't really be a minimum. So in this, when we see a critical point, it's kind of where it's turning around, they're not where it just stops, right? So that will be our critical point for our maximum. Um that we might want to identify, let's do the same thing with a minimum point right here. So I'm thinking you guys can guess where the minimum point is gonna be. Um But let's go ahead and do the same kind of method here. You see that the graph is falling and falling and falling right, and then on the other side it starts rising again. So there had to be a point where it turned around, it was falling for a while then now it's rising, where did it turn around? It's right here, that is our minimum point. And for now, all we wanna do is be able to recognize when a graph has a minimum or a maximum and then later on, we will be able to use this information when we're analyzing graphs. Cool, alright, let's move on

7

concept

Calculating Area of a Triangle on Graph

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8m

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Alright. So now we're gonna learn how to calculate the area of a triangle on the graph. Um We're gonna do this a few different ways in this class but the formula stays the same. You might remember this area of a triangle equals half times base times height. Uh We usually just write this as half B. H. Right? That's the same thing. Half be H. Um So that is our area of a triangle formula. Let's go ahead and use it on the graph a few different ways. So here on graph a. Um what I wanna do is I'm going to pick this point right here and I want to calculate the area um below the blue line but above that point. All right. So that sounds a little crazy but let's go ahead and visualize it on the graph. It's gonna be above or below the blue line and above that point. So what we would have is this line here connecting this to the point. And now you can kind of see what triangle I might be talking about. I'll highlighted here in yellow. Alright. So there's gonna be a few times in this class that will use a graph like this to calculate this area. Alright. So how do we do it first we have to define what's gonna be our base and what's gonna be our height and then we have to find out what those are. Right? So here we're gonna have our base B. This part right here, the dotted line and our height is gonna be coming up right here. That's gonna be our height. So I'll put an H. I'll put an H out here and I'll leave that little squiggly thing. Okay, so that's gonna be our height. Now. What are those numbers? We gotta figure that out. So for our base let's see what it is. We need to find basically what the change in that X value is. So it looks like we started here at zero for the X values, right? Zero down here and we went all the way to three, right from 0 to 3. So the change there is gonna be three, right? We changed three. So the length of that segment is three, let's do the same thing for the height looks like we started at three and we went up to six. So what's the change there? The six minus the three is gonna tell us that our height is three. Whoops, let me draw that a little better. So our height is equal to three, our base is equal to three, our height is equal to three. So we are ready to do our formula, I'm gonna do up here. Um A for area A for area equals a half times base times height. Our base was three. Our height was three, so half times three times three. That's going to be nine over to right and that simplifies to 4.5 as well. Okay so we are going to do a fractions. Review a decimal. Review all this stuff. If any of this math is tripping you up, we have reviews for all of it. All right, so we're gonna go through all that stuff. Let's go ahead and do um example be here. I'm going to get out of the way and let's do something similar. We've got that same point there where they're intersecting. Um But now I want to go ahead and find the area below the this this dotted line. Okay, so now before we did the area above now we want to do the area below. So again I'll highlighted in yellow here, the area I'm talking about. Right? So this area um highlighted in yellow, this triangle, how do we figure out what the area is? So again we're gonna use our same formula a half base times height. So we need to know what our bases and what our height is. So here we've got a base along this dotted line. Again, the base is going to be there, okay and our height is going to be this change in the y over here, so I'll put it out here. H um And again let me highlight what the height is gonna be. It's gonna be this region right there. It's gonna signify the height and I'll do one over here because I like drawing these squiggly, that's going to be our base. Cool. So let's go ahead and calculate what those are. So it looks like for our base we started here again at zero, right? And it went all the way to four. So from 0 to 4, the change four minus zero, the base is going to have four units. How about the height? Well it looks like we started here at one and it went up to four. So four minus one, that's gonna give us a height of three. So here our base was four, our height was three. We are ready to calculate the area area equals half base times height. So we've got half um times our base of four. Our height of three which is gonna be 12/2, it's going to equal six. Okay so don't get caught up on the math. The idea is how do we use this formula one more example here example c. I've got space on the right so I'm coming back. Hey guys. Alright so example see in this situation I want to find the area of this triangle right here. So they could give us this point right here, they could tell us that the X. Value is two and we need to find the area in between this point. So this one seems a little trickier, right? At least a little bit. How do we calculate this this area right here. Alright so though it seems a little trickier, it is almost the same. Um We gotta find the height. So if you look uh kind of sideways at this triangle, you'll see that we've got the height right here along the middle. You see this this line I just drew in black, That's actually gonna be our height right there. That's the height of that triangle. And our base is gonna be this whole long line right here that I'm gonna do I'll do in in green. So this whole long line right here connecting those two points, that is our base. Cool, so triangle looks a little trickier, but in the end, once we define our base, define our height, the math gets a little easy again. So let's go ahead and see what um what the change in our base and what the change in our height is. And honestly you could you could um do it in any order, right? So let's just start with the base and let's see what the change was. So we started here at zero right zero on the Y axis. And it went all the way up to six. So six minus zero. Our base is going to be six, so base equals six. How about our height? It looks like our height. We started here at two and we went to four. So four minus two. Right? We started their height ends, their four minus two is equal to two. So our height is going to equal to. So now that we know are basis six, our height is two. Let's go ahead and calculate our area area equals half be H. So half times the base of six times the height of two. We've got 12/2. Came out to six again, so the area of that yellow region is six. Cool. That's three different ways that we're gonna use this formula in this class to calculate areas of triangle. Alright, let's move on.

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Calculating Area of a Rectangle on Graph

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Alright. So now we're going to calculate the area of a rectangle. This will be similar to how we're calculating area of a triangle on a graph. Sometimes we have to calculate the area of a rectangle. So if you remember from geometry area of a rectangle is your length times your width. You can keep it kind of similar to to our triangle formula and just the base times the height. Right? Um Either way you remember it um It's it's pretty simple formula. So let's go ahead and do an example here. So what I wanna do is I want to find the area of this rectangle. I'm going to to highlight on the graph right now. So what we have is they could give you two points like this and they might ask to calculate what is this area right here. Alright, I'm gonna highlight it in yellow. Just like I've been doing right. How do we calculate that area? So we just have to define the length and the width or the base? And the height. I'm gonna use base and height just to keep it consistent with the triangle videos. So here um the base will be our horizontal portion and our height. We will do as this vertical portion out here. Right? So this will be the height and this will be the base. Let me make a little more space there. This will be the base. Alright so base. Um So let's go ahead and find what the base and the height are. Start with the base. We started with this valley right here which was zero on the X axis, right? And it looks like it went all the way to two. So from two from 0 to 2 it was a change of two to minus zero is two. Let's see what this height is. Looks like. We started at six or and went to three or started at three and went to six. Right. The movement there six minus three. It's gonna give us a height of three. So let's go ahead and calculate the area area equals base times height for a rectangle. And it's gonna be two times three which equals six. Area of that rectangle is six. Alright, let's move on. I've got a practice problem for you for this.

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Problem

Calculate the area of the shaded region.

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Interpreting Graphs (Part One)

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Alright, so now let's do a quick review on interpreting graphs, right? We've been making them up to this point. Now let's analyze them a little bit. So first let's define these two points. These two terms, we've got correlation which is the relationship between two variables that allows us to predict outcomes, right? Um So things are thought to be correlated if we are able to make predictions based on this information and our next uh term here causation, it's a relationship where one event triggers another one. So this is one thing causes another thing. This is basically cause and effect relationship. Right? So causation cause and effect. So let's look at an example here, we've got a graph here with outside temperature on the X axis and ice cream sales on the Y axis. So what I'm trying to point out is that as the temperatures rise, people are gonna buy more ice cream. So we might see a graph something like this, right? Where as the temperature is rising. So are the sales of ice cream? Cool. Um And the idea here is right, we see outside temperature going up, sales going up this relationship. What we see when they go up together or down together. This is called a positive correlation, positive correlation. Um And it's also sometimes called a direct relationship. So this is when we see something like the X values going up, then the Y value is also going up or when the X value goes down, same thing. The Y value is also going to go down, right? So up together or down together is a positive relationship compared to what we call a negative relationship or an inverse relationship. Um That's when they move opposite. So that would be something where we see the X value going up, keep it consistent with the colors there. We'll see the Wye Valley going down and the opposite, right? X going down and why going up? So, let's think of an example of and uh negative or an inverse relationship. Let me get out of the way. We'll put a little graph right here. So, maybe a negative relationship might be something. Uh let's do a little one. Maybe we've got, you know, uh number of miss classes or let's say, absences over here, absence from class. And over here we'll put your grade, right? So the idea is while absences are low, so if you've got zero absences, you might have a really high grade. And as the absences go up, your grade falls. Right. So this is a negative relationship. The absences are going up and your grade is going down. Cool. So now, in the next video, we'll do a little more discussion about interpreting graphs and some of the pitfalls that you might run into

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Interpreting Graphs (Part Two)

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Alright. So now let's discuss some of the problems we might run into when interpreting graphs. So let's look at this left graph first. We've got wages and education. So education on our X axis and wages on our Y axis. Um And you might expect to see something like this where as education goes up so do our wages, right? That's probably why a lot of you are studying right now. Um And the idea is that yeah, your wages will go up in the future as you are more educated. Cool. But what are we missing here? Right. There's another factor to the compensation equation that we might be leaving out. Um So the idea here is that sometimes a graph might omit a variable. So we call this the omitted variable bias. Alright, omitted variable. Um And the idea here is that although education is important for your for your to determine your wage. So is um your experience. Right? So experience in this case is going to be our omitted variable. Right? I would imagine that there is some correlation between the amount of experience you have and what your wage is gonna be. Alright. So that is one way that a graph can omit some information. Right? We're emitting a variable here. Um It's not showing us the full picture. I'm going to get out of the picture now to use this right graph to explain what we call reverse causality, reverse causality. So remember causation is where one thing uh One thing comes before the other right? It's a cause and effect relationship. So reverse causality you can imagine is where you take the effect and you think that the effect causes the cause, right? You're looking at it backwards, not the cause causing the effect. Where you're looking at the effect causing the cause. So it's reverse causality. So the idea here is something like this where we have police officers on the X axis and crime on the Y axis. And the idea here is that it's saying that as police officers increase in the city, so does the crime, right? And that seems kind of backwards, Right? So the idea is like you look at a city with a lot of crime and you're like, hey there's a lot of police officers in that city. So since there's a lot of police officers, that must be why there's a lot of crime um instead of thinking of it the other way around, right? So a city with a lot of crime has a lot of police officers, so they're kind of mixing up the variables here. The idea being that the graph is showing that um police officers cause crime rather than crime causing police officers. Cool. So those are two types of pitfalls that we might run into an omitted variable and reverse causality. Cool, so let's move on to the next video