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Symmetry of Polar Graphs - Problem 3

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s test the symmetry of three equations. Each of these equations a, b and c, represents a polar graph and I want to find out what kind of symmetry these graphs are going to have.

The way I check for symmetry is first I try plugging in negative theta. So I take the 2 cosine theta, plug negative theta and I get 2 cosine negative theta plus 2. Here I use the fact that cosine is an even function. So cosine of negative theta’s the same as cosine theta; that’s 2 cosine theta plus 2. But that’s exactly r. And so, what I just found is that, negative theta r is in the graph. That means that this graph is symmetric about the x axis.

What about this one? I can plug negative theta in, I get 1 plus 2 sine negative theta. Sine is an odd function so this becomes 1 minus 2 sine theta. The minus sign can be pulled out, but this is neither plus nor minus r. So I actually can’t tell if this graph is symmetrical. It’s not automatically not symmetrical about the x or y axis.

Let’s try another test for symmetry. The test for symmetry about the y axis, where we plug pi minus theta. So I’ll take 1 plus 2 sine of pi minus theta. Remember, the sine of pi minus theta is exactly the same as the sine of theta. So this is 1 plus 2, sine theta, which is exactly r. This test works, because we found that r pi minus theta is in the graph, and this is the reflection of r theta about the y axis. So this graph is symmetric about the y axis.

What’s interesting is, the first test didn’t reveal that, so we needed to use the second test. This is why it’s good that we have two tests. Now let’s take a look at the third example. Let me plug in negative theta, I get 8 sine of -2 theta. Again sine is an odd function so I can pull the minus sign out. I get -8 sine 2 theta, which is exactly negative r.

Now that I have that negative r, negative theta, is in the graph, that point is a reflection of r theta around the y axis. This indicates symmetry about the y axis.

What’s really interesting about this function, is that the test doesn’t indicate symmetry about the x axis, but it also has that. So just remember that sometimes, the symmetry tests will not tell you about symmetry that actually exists. So that’s one of the things about polar graphs that you can look forward to. When you’re graphing them sometimes you’ll get unexpected symmetry. But make sure you test for symmetry because if you find that the graph is symmetrical, it cuts down on the number of points you have to plot.

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