Graphing polar functions
The narrator says pi/3 is exactly halfway between pi/4 and pi/2, but halfway is 3pi/8. This is minor and a note corrects it.
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The video graphs the polar function r = 4cos(theta) - 2 by making a table of theta and r values and plotting them on polar graph paper. Many r values are negative, so those points are plotted backwards through the origin. Extra points at pi/4, 3pi/4 and 5pi/4, and the zero at theta = pi/3, help reveal a looped, symmetric curve.
You'll be able to
- build a table of values for a polar function r = f(theta).
- plot points with negative r by moving the opposite way from the angle.
- find where a polar graph passes through the origin by solving r = 0.
- connect plotted points in order of increasing theta to sketch a polar graph.
Key words
- polar function
- A rule r = f(theta) that gives a distance r for each angle theta.
- polar graph paper
- Graph paper with circles for distance and lines for angles, used to plot polar points.
- negative radius
- When r is negative, you go the opposite way from where the angle points.
- symmetric
- Looks the same when reflected; here the graph is mirror-symmetric across the horizontal axis.
Transcript
- [Instructor] We're told to graph the function r is equal to f of theta, where f of theta is equal to 4 cosine of theta minus 2. Okay, I have some nice polar graph paper here to do it. So let me just set up a table and find some values for theta and r, and maybe I can get a rough sense of this graph.
So theta in this column and r is equal to f of theta in that column. So, I could just start at, right at the angles that hit the axes. So I could have theta is equal to 0 radians, and then maybe I'll go to pi/2. Let me go to pi/2. Then I could go to, I could go to pi, then I could go to 3pi/2, 3pi/2, and then I could go all the way back to 2pi.
2pi, let's see what I get. And maybe I'll color-code this so I can keep track of the points. So, when theta is 0, 4 cosine of 0 is, I can just write it over here. This is going to be 4 times cosine of 0 minus 2, cosine of 0 is 1. So it's just 4 minus 2, this is equal to 2.
So when our angle is 0, we're pointing in this direction, we're gonna have a radius of 2, so we'll go out 1, 2. So that is that point right over there. Now let's go to this next point. I'll do this in this teal, light blue, green-blue color. When we're at pi/2, then r is going to equal to 4 times cosine of pi/2, I know that's not super easy to see, minus 2.
Cosine of pi/2 is just 0, so that's all 0. So we're just left with a -2, r is equal to -2. So when we're pointing in this direction, when we have a negative radius, even though we're pointing straight up, because it's a negative radius, we're going to go back to 1, 2.
We are going to end up right over there. And then, let me do this next point. When theta is pi, so we're pointing due left, then we're going to have 4 cosine of pi minus 2. Cosine of pi is -1. So it's going to be -1 times 4, which is -4, this is -1. So it's gonna be -4 minus 2, which is equal to -6.
So we're going in the pi direction, our angle is pi, so we're going straight left, but our radius is -6. So we're gonna go back 6, 1, 2, 3, 4, 5, and 6, this is interesting. And then let's go to 3pi/2. Let me do that in green. So, if I were to say 4 times cosine of 3pi/2 minus 2, cosine, we're going straight down, cosine of 3pi/2 is 0.
So this gets us to -2 again. So if we're going straight down, but our radius is -2, that means we're going to go backwards 2. So that's 1, 2. So we're gonna go straight up 2. So that's that point right over there. Oh, this is, I'm having trouble visualizing it just yet, but let's see if we can make some progress.
Let's switch over to pink. At 2pi we're gonna have 4 times cosine of 2pi minus 2. Cosine of 2pi is 1. And so 4 times 1 minus 2, we are back to 2. So then we are back to this point right over here. So we've made a full circle, although I don't think this is going to be a circle, but we've gotten back to where we started.
But the, if I were to try to connect these dots, it would look something, something like, maybe something like that. Remember, I'm trying to connect them in the order that, as I increased the angle, so it might look something like that. It's probably symmetric.
But just to make sure, let me actually do some more points. Let me do the points that are halfway between the various, or halfway through each quadrant. So let me, let me find some new colors here. So let's try pi/4, which would be that angle. Then we could do, we could do, what would it be, 3pi/4, which would get us to that angle.
3pi/4, which would get us over there. And let's see if that can help us out. I could do more if I need it. I could also do this angle right over here, which would be 5pi/4, 5pi/4, maybe, maybe that's sufficient. So for pi/4 we're gonna have 4 times cosine of pi/4 minus 2.
Here we're gonna have 4 times cosine of 3pi/4 minus 2. Here we're gonna have 4 times cosine of 5pi/4 minus 2. Now, cosine of pi/4 is square root of 2 over 2. So this is all going to simplify to 2 square roots of 2 minus 2. If I do 4 times 2 square roots of, or if I do 4 times square root of 2 over 2, I'm going to get 2 square roots of 2.
So that's what I get over here, and I'm gonna use a calculator to evaluate that. Cosine of 3pi/4, when I'm over here, is negative square root of, negative square root of 2 over 2. So this one is going to simplify to negative 2 times the square root of 2 minus 2.
This one over here, 5pi/4, is actually going to have that same value. It's going to be the same value as we have over here, 'cause cosine of 5pi/4 is the same as cosine of 3pi/4. Our x-axis, or x-coordinate's going to be the same in either situation. And, well, actually, that's going to be, hopefully maybe that's sufficient.
Let me get the calculator out and calculate these. So, if I take... 2 times 2 times the square root of 2, that is equal to that. So it's roughly equal to 2.8, and we'll just see, to get an approximation, to see if we can sketch that. So if... this is... approximately 2.8, then this is approximately, this is approximately going to be equal to 0.8.
So we have the angle pi/4, that's the angle, and we're only gonna go 0.8. So we're gonna go right about there. So we're gonna go from that first point to the second point. So the curve will look something, something like this. The curve is gonna look something like that, maybe.
And then, from there we now go to this blue point, so maybe it goes like this. It maybe goes something like that. And we could see, well, this, is there a point where this would equal 0? Yes, there actually is a point where this could equal 0. You can definitely find a theta that is going to set that equal to 0.
Actually, I think it's gonna be when theta is pi/3, we're going to hit this point right over there. And pi/3 is exactly in between pi/4 and pi/2. So yeah, the next point would be you'd go through there, then you'd go down here. And now we would go to 3pi/4, which is this angle.
And at 3pi/4, you have this negative... 2.8 minus 2, which would be -4.8. So approximately -4.8. So 3pi/4, you're gonna go backwards. So 1, 2, 3, 4, 0.8, you're gonna end up someplace around there. So the next point is gonna look something like this, and then you're gonna come around, and you're gonna come around and go up, do something like that.
And if we assume some symmetry, and I actually think there will be based on how we calculate it, then we could, then it's the graph, and you can verify this, probably is going to look something, something like this. Going back down there, going through that point, and then getting back to where we started.
I encourage you, well, hopefully you understand everything we just did, and also go on to some graphing calculator and try to graph it yourself and see if we got pretty close.
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