Describing characteristics of polar function graphs

0:00- [Instructor] Let g of theta be equal to all of this. Select the dropdowns that best describe r equals g of theta on the interval from pi over two to three pi over two, and we have these less than or equals here, so it's including those two endpoints. So as theta increases from pi over two to three pi over two, the value of r is, and this I copied and pasted this from Khan Academy, but this dropdown says positive and increasing, positive and decreasing, negative and increasing, negative and decreasing.

0:28And then we have to pick whether it's increasing or decreasing the distance from the polar point to the origin. So like always, pause this video, have a go at that, and we'll do it together. Alright, well one way to do this is just to try to set up a table here and just think about what's happening as theta goes from pi over two to three pi over two.

0:47So let's do theta right over here and let's do g of theta right over there. And the graph is gonna be r is equal to g of theta. So we're gonna start at pi over two, and I'm just gonna find some values that would be easy to evaluate this. So maybe we'll go to pi over two and then maybe we'll go to pi and then maybe we'll go to three pi over two.

1:11So when theta is pi, we have... Oh, so when theta is pi over two, pi over two minus pi over two is equal to zero, so it's gonna be cosine of a zero, which is one minus one, which is zero. Alright, when theta is pi, pi minus pi over two is pi over two. Cosine of pi over two is zero minus one; gets us to negative one.

1:41And then last but not least, when we get to three pi over two, you subtract pi over two, and it essentially you get to pi. Cosine of pi is negative one; minus one is negative two. So as theta increases from pi over two to three pi over two, the value of r is, well, we could see it's non-positive the entire time.

2:10Right at pi over two, we are at zero and then it's negative after that. So we're getting more negative and we are decreasing. The magnitude is increasing, but the actual value is decreasing, so it is negative and decreasing. And I'm assuming they say right as we leave pi over two because at pi over two, we're not quite negative.

2:34We are, I guess we could say we're... I would say we are non-positive and decreasing if we wanted to include pi over two in our interval. And let's see, therefore the distance from the polar point to the origin is... So this is interesting because we could actually plot these points; that might be the easiest thing to see what's going on.

2:57Because if we think about it this way, let me get another color here. So this angle is pi over two and r is zero. So when the angle is pi over two, you're going straight up, but the radius is zero, so you're right over there. And then you go to... Let me do another color here.

3:16Then you go right over here. When our angle is pi, we're going straight to the left, but our radius is negative one. If it was one, it would be one unit to the left. (dog barking) Sorry, my dog is barking. But since it's negative one, we would be one unit to the right.

3:30So the next point we get to, if this is one unit to the right, would be right over there. And I don't know if this is some type of a curve, so maybe we are negative... Or, well, let me do the next point and maybe I can make sense of this. So last but not least, three pi over two and negative two.

3:48So three pi over two is straight down, but our angle; our radius I should say; is negative two, so it'd be two steps back, so we're going to be over there someplace. So I don't... I would probably wanna plot more points, but things like this tend to look something like, maybe it's looking something like that.

4:11And so if it does look something like that, if that's roughly what the curve is looking like, then the distance from the polar point to the origin, so the point on the curve to the origin, seems at least over that interval to be increasing. Increasing. And the big takeaway here is if my radius is negative and it's becoming more negative, why is it increasing?

4:37Well, when you have both of those happening and actually... Well, actually, yes, when you have both of those happening, when you have negative radii, and you're getting more and more negative, whatever your angle is, you're getting further and further in the opposite direction away from the origin, and so that's why the distance from the polar point to the origin is increasing.

4:58Another way to think about it is the distance from the polar point to the origin is just the absolute value of these, and that is increasing.