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Symmetry of Polar Graphs - Problem 1

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.

How do you recognize symmetry across the y axis of a polar graph? Well, I’ve drawn a picture of some points plotted in the polar coordinate system and I have point P here. And I also plotted points Q and T. Notice that Q is the reflection of point P across the x axis and I get that point just by switching theta to negative theta.

Now this point, point T is the reflection of point P in the y axis. To get this point, one way to get it, is to take the opposite of both coordinates. Think of it as, I’m facing in the negative theta direction and I’m walking -r in that direction, so I’m walking backwards r units. And I end you here at point T. So these two points are symmetric about the y axis or the line theta equals pi over 2. This is the basis for one of our tests for symmetry about the y axis.

Here I’ve got a graph of the curve y equals 2 sine theta and it looks like it’s symmetric about the y axis. This is going to give us the basis for our test for symmetry. Now point r theta is one the graph, I’m going to need negative r, negative theta to also be on the graph. So my test is going to be, plug in negative r negative theta and see if the equation’s true.

Let’s take this equation; 2 sine theta, I’m going to plug in negative theta and let’s see if the equation’s true. Now remember the sine is an odd function so the minus sine can be pulled out. Negative sine theta and 2 sine theta is exactly r. So this is minus r.

What I’ve just shown is that when I plug in minus theta, I do get minus r. So minus r minus theta is on the graph and it checks. This equation does represent the graph that’s symmetric around the y axis. The test for symmetry about the y axis, that we’ve just gone over is, replace theta with minus theta and see if you can get negative r.

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