Hey, guys. Hopefully, you gave this one a shot. We're going to get some more practice with position diagrams. Let's take a look at this example together. We have a position diagram for a moving car, with points a through e. Let's start off with the first one. Where is the car moving the fastest? So again, we'll just stick to the steps. Which variable are we talking about? Position, velocity, or acceleration? Well, we're talking about motion here. Where are we moving the fastest? So it's not going to be the position. And remember, the acceleration is a change in motion. So we're actually not going to talk about acceleration either. It's going to be the velocity. That's the first step. Now, the graph feature, we just get by using the table down here. When we are looking at a position-time diagram, we are looking for the velocity. That means we're going to look at the slopes of the graph. There are a couple of rules to remember: upward slopes are going to be positive, flat is 0, and downward is negative. And the steeper you are, the faster you're going. That's the rule we're going to use. We are looking for the slopes here. The third step is the qualifier. In which of these terms over here in this list are we going to use? Well, the fastest is the keyword here, and the fastest means like the highest velocity. In this list, that's actually going to be the maximum value over here. We figured out that we're looking for the maximum slope. Notice how the question also doesn't mention forwards or anything like that. So we're just looking for the most vertical slope. Now, we just have to interpret it from a graph. We have to draw out the slopes for each of these points by looking at the tangent lines. Those are going to be the slopes for each of these points. Now I just have to figure out which one of these points is the steepest, and if we take a look, that's actually going to be point c. So that's our answer.

Moving on to the second one. Where is the car moving the slowest? Again, guys, if you go through all the steps, where moving the slowest means we're going to talk about velocity. We know that means we're looking at the slopes. So now for the qualifier. Well, if the fastest meant we were looking at the maximum, then the slowest means we're just going to look for the minimum. And if the steepest slope was the maximum, the flattest slope is going to be the minimum. So let's look at the graph. Where do we have flat slopes? Well, it's actually pretty easy. At points b and d, we have flat slopes. So those are going to be points b and d. Those actually also are where you have no motion at all. It's v equals 0. So if you're not moving, that's the slowest you go.

Now, move on to part 3. Where is the car turning around? What does turning around mean? Well, if you turn around, you're going to be moving in one direction, you stop, and then you move in the other direction. So we're still talking about motion here, which variable are we going to use? We're going to use the velocity. So we're talking about velocity, which means we're going to talk about the slope. Those are the first two steps. Now, we just have to look at the qualifier. Remember what turning around means. It means you're moving in one direction, and then you turn around and move in the opposite direction. You're talking about a change in direction. And so remember, when you're talking about the slopes, when you have upward slopes, that's going to be positive velocity. That means you're moving forward. Downward slopes are going to be negative velocity, which means you're moving backwards. So when you're looking for a direction change, you're actually looking for a sign change. At point a, because it's upward, you have a positive velocity. Point b, you have v equals 0, and then point c, because it's downward, you have negative velocity. Which means somewhere inside of this little point over here, you were moving forward, you stopped, and then you turned around and started moving backward. That actually happens right here at point b. It happens when you stop momentarily and then right before you start moving in the opposite direction. So that means that happens at point b. There is no other point with a direction change, so point d is not a turning point here.

Last two parts, where is the car speeding up and slowing down? Let's just go through our variables. Which are we talking about? Position, velocity, or acceleration? Well, you might think, oh, we're talking about speed, so we're going to be looking at speed. But actually, what's happening is speeding up means the speed is increasing. It's talking about a change in that speed. So we're actually going to be talking about the acceleration here. Now we just have to figure out which graph feature we're talking about. Again, going down to our table, we're going to see that when we talk about the acceleration in an x-t graph, we're looking at the curvature. There are a couple of rules to remember. When we have a smiley face upwards like this, that's going to be positive acceleration. A frowny face is going to be negative acceleration. We have to remember that if you break up these smiley faces and frowny faces into two halves, then on the left side, you're always going to be slowing down. The right side, your slopes are becoming more vertical here, so you're actually going to be speeding up. So let's go back to the question. We're going to be looking at the curvature over here for speeding up for the acceleration. So which qualifie...