Graphing Review

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.

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.

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