Graphing Review - Online Tutor, Practice Problems & Exam Prep

To calculate the average slope of a curve, use the arc method by selecting two points and drawing a line between them. The slope is determined by the formula: $\mathrm{slope}=\frac{\mathrm{rise}}{\mathrm{run}}$, where rise is the vertical change and run is the horizontal change. For area calculations, the area of a rectangle is found using the formula: $\mathrm{Area}=\mathrm{base}\mathrm{height}$. Understanding omitted variable bias and reverse causality is crucial for accurate graph interpretation.

Calculating Slope of a Curve:Arc Method

Alright. So now we're going to calculate the slope of a curve that's not a straight line, using the arc method. 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. 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. 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. 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,m=riserun, so the slope of that line connecting that arc is going to equal our rise over our run. Our rise was 1, our run is 3, our slope is one 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

Calculating Area of a Rectangle on Graph

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, 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. What I want to do is to find the area of this rectangle. I'm going to highlight it on the graph right now. So, they could give you two points like this, and they might ask to calculate what is this area 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. It 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⋅3=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.

Alright. So let's solve this problem. Calculate the area of the shaded region. We are talking about this square, which is also a rectangle right here. We're going to use our formula area equals base times height or length times width, however you remember it. I'm going to use base times height to find the area of that region. Alright, so let's start with the base. I'm going to call this right here the base. From there to there, that will be our base, and we will say from here to here is our height. Alright, so let's find out what that base is and what that height is in numbers. It looks like for our base, we started here with an x-value of 0 and got to a value of 3. So 3-0 is going to tell us that our base is 3, and let's do the same thing with height. It looks like the height was at 2, went up to 5, so 5-2 tells us that the height is 3. So the area is going to be the base times the height which is 3×3. Our area is 9. Cool.

Interpreting Graphs (Part Two)

Alright. So now let's discuss some of the problems we might run into when interpreting graphs. Let's look at this left graph first. We've got wages and education, so education is 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 yes, 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? This omitted variable, and the idea here is while 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. 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 are a lot of police officers in that city. So since there are 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.

Here’s what students ask on this topic:

How do you calculate the average slope of a curve using the arc method?

To calculate the average slope of a curve using the arc method, select two points on the curve and draw a straight line connecting them. The slope of this line represents the average slope over that section of the curve. Use the formula:

$\mathrm{slope}=\frac{\mathrm{rise}}{\mathrm{run}}$

where 'rise' is the vertical change between the two points and 'run' is the horizontal change. For example, if the rise is 1 and the run is 3, the slope is:

$1/3$

What is the formula for calculating the area of a rectangle on a graph?

The formula for calculating the area of a rectangle on a graph is:

$\mathrm{Area}=\mathrm{base}\mathrm{height}$

To find the base and height, identify the horizontal and vertical distances between the two points defining the rectangle. For example, if the base is 2 units and the height is 3 units, the area is:

$23=6$

What is omitted variable bias in graph interpretation?

Omitted variable bias occurs when a graph leaves out a variable that significantly affects the relationship being studied. For example, a graph showing the relationship between education and wages might omit the variable 'experience,' which also impacts wages. This omission can lead to incorrect conclusions about the relationship between the variables shown.

What is reverse causality in graph interpretation?

Reverse causality occurs when the direction of cause and effect is misunderstood. For example, a graph might show that cities with more police officers have higher crime rates, suggesting that police presence causes crime. In reality, higher crime rates likely lead to more police officers being deployed. Understanding the correct direction of causality is crucial for accurate graph interpretation.

How do you calculate the slope of a line on a graph?

To calculate the slope of a line on a graph, use the formula:

$\mathrm{slope}=\frac{\mathrm{rise}}{\mathrm{run}}$

'Rise' is the vertical change between two points on the line, and 'run' is the horizontal change. For example, if the rise is 4 units and the run is 2 units, the slope is:

$4/2=2$