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 2 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 going to label what are called the axes 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 going to have numbers or values that tell us how much each jump in the graph is, and for now we're just going to go 1,2,3,4,5,6,7. Right? And usually when I give you graphs in this class, you're not going to 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 the demand schedule, and it's got some prices and some quantities. Right? So at certain prices, there's going to be certain quantities that are demanded, and later in the course, we're going to dive into these topics more. But for now, what I want to do is get these points onto the graph. Right? So the first thing we have to do is we're going to label one of our axes as the price and one of our axes as 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 going to 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 (6, 1), a price of 6 and one demanded. So I'm going to go to my price axis and find 6 right up here, and I'm going to start going out. Right? And then when I go to the quantity, I'm going to find 1 and I'm going to go up from there and I'm going to find the point where these two cross with each other, right? So right here that's going to end up being the point 6 for price and one for quantity right there. I'm going to erase some of those extra. Cool? So that's going to be point a right there. Let's go ahead. I'm not going to change colors because I'm not going to have 5 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 red just, just to keep it a little consistent. So, here we have a price of 5 and a quantity of 2. So we'll find 5 on our price axis, 2 on our quantity axis, and we'll find the place where they mix or where they meet, and that's going to 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 4 and a quantity of 3. Right? And that'll be point c right there. And now point d, I'm going to do in blue. And that's got a price of 3 and a quantity of 4. So you can see these are kind of the places where you could get mixed up. Which way, you know, which axis do I put the 3? Which one do I put the 4? So you just have to make sure that you're on the right axis when you start counting. So that'll be point d right there, and let's finish it up with e at a price of 2 and a quantity of 5. Right there. That's point e 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.

# Graphing Review - Online Tutor, Practice Problems & Exam Prep

### Plotting Points on a Graph

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### Drawing and Shifting Curves

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So now we're going to 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 going to learn how to shift curves just visually with no math. It's a tool we'll use quite often in this class. So let's go ahead and look at 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, what we do is we start at the leftmost point and work our way rightward. Okay. This one seems pretty simple. It's just going to 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. I just want to make an example here. I'm going to 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. You never want to double back and start going back to the left or back to the right. Let me show you 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. You start at the leftmost point and 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, 2 units to the right. 2 units to the right. Okay? So how do we do that? What we're going to do, and the easiest way I find to do it, is I pick the leftmost point, so in this case, it would be this point that I'm going to circle here in black. Right? And we're going to move it 2 spaces to the right, so I'm going to count here 2. That's 1 and that's 2 right there. That's going to be our new point that I'll put in blue. Cool? So you do that with your leftmost point and I like to just go straight to the rightmost point and do the same thing. Grab my pen and I'll pick this rightmost point right down here, and I'm going to move it 2 spaces to the right. 1, 2. Right? And I'm going to put my point right there, my new point, and now that I have two points, 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 2 to the right. Actually, I'm going to do it in blue so we can see which one's which. So, the blue curve has been shifted 2 to the right. Cool? And a lot of times in this class, like I said, we're going to be doing the shifting of curves just visually. We're not going to put any math behind it. We're going to have a reason we're shifting the curve, and then we're going to have to see what happens after we've shifted the curve. And when I say see what happens, we're going to see what happened to the new price and the new quantity. But we'll get more into that in the next chapter.

So I'm going to draw a couple 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 going to kind of draw a graph kind of willy nilly like this, and a lot of them are going to be graphs that look like this. We're going to have an X and remember I suggested having at least 2 colors, and we're going to use those quite often. So in this case what we're going to do is kind of like we did above, on the graph. We're going to shift the red line, to the right. So now it's not 2 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 going to 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. 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.

### Calculating Slope of a Straight Line

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Alright. So let's continue by learning how to calculate the slope of a line. You might remember this from algebra. Right? So here in the green box, I've got our formula for slope. It's going to be our rise over run, the change in y over the change in x, y2−y1÷x2−x1. These all mean the same thing. In this class, it's probably not going to 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.

Let's start with this first graph, Graph A, with the red line. My first step when calculating slope on a graph is to find two points that intersect on the graph directly, one of these intersections on the graph. This first graph actually has quite a few of them. So, you know, right here, right here, right here. Right? Some of our other, I'm going to go ahead and pick this point and this point. Two points that intersect the line there, and let's go ahead and calculate the rise first and then we'll do the run.

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 how much the up and down changed between these two points. So we started here at 5, and it looks like the next point is down here at 4, right? So it looks like we went down 1, and when we're calculating slope, down is going to be negative and up is positive. 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 and to the right is positive. Cool. So in our example here, like I said, we went down 1, so that is going to be 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 to the 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 1. Our run was 1, so that's going to simplify to negative one. Our slope here is negative 1. If you need a little refresher with fractions as well, I'm also including a fractions review in this section too. Cool. So let's move on to part B here, and let's calculate the slope here.

I'm going to get out of the way so we can see the example, and let's remember, like I said, the first step, we want to find 2 points that are intersecting the graph, at one of those intersections. Right? So you can see 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 going to 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 2, and the next point is at a vertical value of 3. So let's see. We're going to draw our arrow here, and it looks like it went up 1, from 2 to 3. So I will write 1 right here, and now let's do our run. So we started at 3 and we went to 6, so it looks like our change was 3 here, right? From 3 to 6, our run is 3. 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 1, our run was 3, and that's it. The answer is 1 third. The slope of this line is 1 third.

So let's scroll down here. 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 2 points, where it's intersecting directly 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 it 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 4. Our next vertical value is 6. So it looks like we're going to go up here, and it looks like we went up 2. Right? We started at 4, went to 6. So our rise was 2. Let's do the same thing with our run. In this case, it looks like we started at 3 and we got to 4. So it looks like our run is going to be 1 in this case, and let's calculate the slope. So our slope, again, rise over run. Right? And our rise was 2. Our run was 1. 2 over 1, that's just 2. So our slope in this case is 2.

So let's go ahead and compare, 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, this line is going downhill, right, because the slope is negative, it forms a downhill, going from left to right. And notice our other 2 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 in B, our slope was 1 third and in C our slope is 2. Right? So 2 is quite a bit bigger than 1 third and look at how these lines look, right. In B you kinda see like a soft growth here, right. It's kind of a little bit of an uphill whereas in C, where we have a slope of 2, it's a lot steeper. So the higher the slope is, the steeper it's gonna get this way. And if it was a really negative number, so if it was negative 2, you could imagine it'd be a lot steeper going down. Cool? Alright. So that's how we calculate slope. Let's move on.

### Calculating Slope of a Curve:Point Method

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Alright. So now we're going to learn how to calculate the slope of a curve when it's not a straight line. So the idea here on this graph what you see is a curve 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. So you're going to have a different slope at this point where it's rising pretty fast than at this point where it's kind of going up a little slower, right? You would imagine from our last video that those would have 2 different slopes.

So the first method when calculating the slope of a curve is to use what's called the point method, and what we do is we draw a tangent line. Right. 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 going to go ahead and do my best to draw a tangent line. It's not very easy to do this by hand. If you were ever to have to calculate this in this class, I'm sure they would give you the tangent line already, and I'll do my best here. It 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 to go and it touches the graph and it keeps going. Just 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, using our same method from finding the slope, let's find 2 points that intersect the graph, and it'll make it easier to calculate. So I see one there. Here's another one right here. Let's go ahead and calculate that slope. So it looks like from the first point to the second point we are going up. Right. Let me do this in a different color. I'll do it in green. 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 2, and let's do our run now. So it looks like we started with an x-value of 3. We got to an x-value of 5. So our run was also 2.

So let's calculate this slope. So the slope of the tangent line, I'll put riserun, and in this case, we've got a rise of 2, a run of 2, and that simplifies. 22 simplifies to 1. So the slope of the tangent line is 1, and that means that the slope at this point where we drew the tangent line right here, the slope of that point equals 1. So the slope of that curve at that point is 1. Remember it's constantly changing but at that point the slope is 1. So that is how we calculate the slope of a curve, using the point method. Let's move on to the next video.

### Calculating Slope of a Curve:Arc Method

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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 going to find the 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, I've got instructions here on the right, where you want to draw a line connecting the ends of the arc. Right? So, the region that you want to calculate the slope over. Right, I've got 2 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 going to get different answers in all those cases because the slope is constantly changing, and you can pick any points, right. I just picked points where we've got intersections, it's just going to make the math easier. So here, this is the slope, this slope is the average. Average slope over the region, over that arc. So between those two points that you select and calculate, you're going to 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, and we're going to find an average. So what I'm going to do is I'm going to pick these two points right here, and I am 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 the graph is actually doing. Right? So this is why we're kind of finding an average. We're doing our best to estimate what that slope is. So it's almost like we've got a line here going like this, right? We calculated those points and we're going to have this line going like that. So let's go ahead and calculate the slope of that line. So same thing. We're going to 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, excuse me, at 4 and we went up to 5. So it looks like our rise was 1. And let's do the same thing for our run. So we started at 3. It looks like we went over to 6. So it looks like our run was 3. Right? So using our same formula for slope, so the slope of that line connecting that arc, the slope is going to equal our rise over our run. Our rise was 1, our run is 3, our slope is 1 third. So that is the average slope over that arc. Right? And if we had picked 2 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 2 points, draw the line, and calculate the slope. Cool. Let's move on.

### Finding Maximum and Minimum on a Graph

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Alright, so now we're going to 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 in this example, I've got this kind of upside-down U. So, how do we find the maximum or minimum point? Well, you'll see that not all graphs have a maximum or a minimum. 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've got to find that point where it turns around. Notice here it's still rising a little bit. Right? It's still rising a little bit, and then here it's pretty clear that this is the point where it turns around. I'm going to 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 going to want to be able to do this and find maximums and minimums on a graph. So, what you'll notice is this one doesn't have a minimum. It went up, up, up to a point and then started going down, down, down. There wasn't a point where it was at a max bottom or max top. You might think this is a minimum here, this is a minimum here, but usually, these graphs are going to continue. So, it would continue going down, and 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 there, not where it just stops. Right? So that will be our critical point for our maximum, 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 going to be, 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 want to 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.

### Calculating Area of a Triangle on Graph

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Alright. So now we're going to learn how to calculate the area of a triangle on the graph. We're going to 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. We usually just write this as 1 2 × b × h . Right? That's the same thing. 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, I want to calculate the area below the blue line but above that point. Okay, so that sounds a little crazy, but let's go ahead and visualize it on the graph. It's going to 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 highlight it here in yellow. Alright. So there are going to be a few times in this class that we'll use a graph like this to calculate this area. Alright, so how do we do it? First, we have to define what's going to be our base and what's going to be our height, and then we have to find out what those are, right? So here we're going to have our base be this part right here, the dotted line, and our height is going to be coming up right here. I'll put an H out here and I'll leave that little squiggly thing. Okay, so that's going to be our height. Now what are those numbers? We have to 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 0 for the x values, right. From 0 down here and we went all the way to 3, right. From 0 to 3, so the change there is going to be 3, right? We changed 3. So the length of that segment is 3. Let's do the same thing for the height. It looks like we started at 3 and we went up to 6. So what's the change there? The 6 minus the 3 is going to tell us that our height is 3. So our base is equal to 3. Our height is equal to 3. So we are ready to do our formula.

I'm going to calculate the area here: A = 1 2 × b × h = 9 2 , 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.

Let's go ahead and do example B 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, but now I want to go ahead and find the area below this dotted line. Okay, so now before we did the area above, now we want to do the area below. So again I'll highlight it in yellow here, the area I'm talking about. Right, so this area, highlighted in yellow, this triangle, how do we figure out what the area is? So again we're going to use our same formula, half-base times height, so we need to know what our base is and what our height is. So here we've got a base along this dotted line. Again, the base is going to be there and our height is going to be this change in the y over here. So let's go ahead and calculate what those are. So it looks like for our base, we started here again at 0, right, and it went all the way to 4. So from 0 to 4, the change 4 minus 0, the base is going to have 4 units. How about the height? Well, it looks like we started here at 1 and it went up to 4. So 4 minus 1. That's going to give us a height of 3. So here our base was 4, our height was 3. We are ready to calculate the area.

Area equals A = 1 2 × b × h = 12 2 , which equals 6. 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 C. 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, you know, the x value is 2 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 area right here? Alright, so though it seems a little trickier, it is almost the same. We have to find the height. So if you look, kind of sideways at this triangle, you'll see that we've got the height right here along the middle. That's the height of that triangle and our base is going to be this whole long line right here that I'm going to do 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 and define our height, the math gets a little easy again. So let's go ahead and see what the change in our base and what the change in our height is and honestly, you could 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 0. Right? 0 on the y axis and it went all the way up to 6. So 6 minus 0, our base is going to be 6. So base equals 6. How about our height? It looks like our height, we started here at 2 and we went to 4. So 4 minus 2. Right? We started there, height ends there. 4 minus 2 is equal to 2. So our height is going to equal 2. So now that we know our base is 6, our height is 2, let's go ahead and calculate our area.

Area equals 1 2 × b × h = 12 2 , which comes out to 6 again. So the area of that yellow region is 6. Cool. That's 3 different ways that we're going to use this formula in this class to calculate areas of triangles. Alright. Let's move on.

### 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 were calculating the area of a triangle on a graph. Sometimes we have to calculate the area of a rectangle. So if you remember from geometry, the area of a rectangle is your length times your width. You could keep it kind of similar to our triangle formula, just the base times the height. Right? Either way you remember it, it's a pretty simple formula. So let's go ahead and do an example here. So what I want to do is I want to find the area of this rectangle. I'm going to highlight it on the graph right now. So what we have is they could give you 2 points like this, and they might ask to calculate what this area is right here. Alright. I'm going to 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 going to use 'base' and 'height,' just to keep it consistent with the triangle videos. So here, 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, let's go ahead and find what the base and the height are. Start with the base. We started with this value right here which was at 0 on the x-axis, right, and it looks like it went all the way to 2. So from 2 from 0 to 2, it was a change of 2. 2 minus 0 is 2. Let's see what this height is. Looks like we started at 6 or and went to 3 or started at 3 and went to 6. Right. The movement there, 6 minus 3, it's going to give us a height of 3. So let's go ahead and calculate the area. Area equals base times height for a rectangle and it's going to be 2 times 3 which equals 6. The area of that rectangle is 6. Alright. Let's move on. I've got a practice problem for you.

Calculate the area of the shaded region.

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### Interpreting Graphs (Part One)

#### Video transcript

Alright, so now let's do a quick review on interpreting graphs. Alright? 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? So things are thought to be correlated if we are able to make predictions based on this information, and our next term here, causation, it's a relationship where one event triggers another one. So this is one thing causes another thing. This is basically a 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 going to 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. 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, 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, the 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, 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 y value going down and the opposite, right. X going down and y going up. So let's think of an example of a 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, let's do a little one. Maybe we’ve got, you know, number of missed 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 0 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.

### Interpreting Graphs (Part Two)

#### Video transcript

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, 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, 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. So the idea here is that sometimes a graph might omit a variable. So we call this the omitted variable bias, alright? Omitted variable, and the idea here is that although education is important for determining your wage, so is 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 going to be. Alright, so that is one way that a graph can omit some information, right? We're omitting a variable here, 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 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 a 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. 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 police officers cause crime rather than crime causing police officers. Cool. So those are our 2 types of pitfalls that we might run into, an omitted variable and reverse causality. Cool? So let's move on to the next video.