Slope and intercepts from tables

We're told Kaia rode her bicycle toward a tree at a constant speed. The table below shows the relationship between her distance to the tree and how many times her front tire rotated.

So, once her tire rotated four times, she was 22 and a half meters from the tree. Then, when she rotated eight times, she was 12 and a half meters from the tree. When it rotated 12 times, then she was only two and a half meters to the tree. So, she's getting closer and closer the more rotations that her tire has had.

Then they ask us some questions here. They ask us how far away was the tree to begin with, how far does Kaia travel with each rotation, and how many rotations did it take to get to the tree?

So, like always, pause this video and see if you can answer these questions on your own before we do it together.

Let's start with the first question: how far away was the tree to begin with? So, the way that I would think about it is after four rotations, we were 22 and a half meters from the tree. We see that if we increase by another four rotations—so let's see plus four rotations—we see that we have gotten 10 meters closer to the tree, or our distance to the tree has gone down by 10 meters.

So, I'll write negative 10 meters here. If we want to figure out how far the tree was to begin with, we have to go back to zero rotations. So, if we're going back by four—so we're subtracting four from the rotations—and if we're going at a constant rate, well then we would add 10 meters. We would add 10 meters; if we add four rotations, we get 10 meters closer. If we take away four rotations to get us back to zero rotations, then we will go 10 meters further.

So, that would be at 32.5 meters. Thus, 32.5 meters is how far the tree was to begin with when Kaia had zero rotations.

The next question is: how far does Kaia travel with each rotation? Alright, well we already saw that with four rotations, she's traveling 10 meters. So, we could say 10 meters in four rotations.

So if we divide both of these by four, what would we get? Well, that's the same thing as two and a half meters in one rotation. So this is 2.5 meters.

Last but not least, how many rotations did it take to get to the tree? Well, we know that after 12 rotations, she's only two and a half meters away from the tree. We also know that in every one rotation, she gets two and a half meters closer. So she only needs one more rotation to cover this next two and a half meters.

If we go plus one rotation, we're going to go down two and a half meters. We're going to go two and a half meters closer to the tree and we will be at the tree. So, how many rotations did it take to get to the tree? In total, 13 rotations.

Now, one thing that's interesting is to think about what we just did in a graphical context that you might have seen before. If we were to put on the horizontal axis rotations and if we were to put on the vertical axis distance to the tree—distance to the tree—I'll just call the vertical axis the y-axis and the horizontal axis the x-axis.

Well, we could see here that we have 0, 4, 8, 12... I could go to 16. And then we saw that at zero rotations, we are 32.5 meters from the tree. So, 32.5—this is all going to be in meters.

This first question was really another way of asking what is our y-intercept. This next question: how far does Kaia travel with each rotation? Well, we saw that when you increase your rotations by 4, your distance to the tree goes down by 10 meters. So when this is plus 4, we went down—we went down 10 meters, so it's negative 10 meters.

What we were thinking about right here is you could think about the magnitude of the slope. The slope of this line, the slope of this line that would describe her distance to the tree based on the number of rotations, the slope is going to be our change in our distance, which is negative 10 for our change in rotations over 4.

So, the slope of this line is negative 2.5 meters per rotation. But when they say how far does Kaia travel with each rotation, she's getting two and a half meters closer. Her distance goes—her distance from the tree goes down by two and a half meters.

And this last question: how many rotations did it take to get to the tree? Well, at what point is our y-value—our distance to the tree—zero? We saw that it is at 13 rotations. So, this is another way of thinking about what was the x-intercept.

The line—and it's a line because we know that she's traveling at a constant rate—looks something like that. So, they really were asking us the y-intercept, the slope, and the x-intercept.