Symmetry tests for polar coordinates 1. Replace θ with -θ. If an equivalent equation results, the graph is symmetric with respect to the polar axis. 2. Replace θ with -θ and r with -r. If an equivalent equation results, the graph is symmetric with respect to θ = 2π. 3. Replace r with -r. If an equivalent equation results, the graph is symmetric with respect to the pole. Note: It is possible for a polar equation to fail a test and still exhibit that type of symmetry when you finish graphing the function over a full period. When you started to graph functions (in rectangular form) you stared by learning the basic shapes of certain functions such as lines, parabolas, circles, square roots, and absolute value functions just to name a few. Polar equations also have some general types of equations. Learning to recognize the formulas of these equations will help in sketching the graphs.
The Correct Answer and Explanation is:
In polar coordinates, symmetry tests are useful tools for understanding the graph of a given equation. Let’s break down the three main tests for symmetry:
- Symmetry with respect to the polar axis:
The polar axis is essentially the horizontal axis in polar coordinates, and it corresponds to the line θ = 0 (or the positive x-axis in rectangular coordinates). To check for symmetry with respect to the polar axis, replace θ\thetaθ with −θ-\theta−θ in the polar equation. If the equation still holds true, the graph is symmetric with respect to the polar axis. This is because replacing θ\thetaθ with −θ-\theta−θ essentially reflects the graph across the polar axis, and if the equation remains the same, the reflection does not change the shape of the graph. - Symmetry with respect to θ = 2π:
The line θ = 2π is essentially the line where one complete revolution around the origin in polar coordinates brings you back to the starting point (this is the same as θ = 0, but after a full rotation). To test for symmetry about this line, replace θ\thetaθ with −θ-\theta−θ and rrr with −r-r−r. If the equation remains the same after this replacement, the graph is symmetric with respect to θ = 2π. This test is often useful for detecting symmetry in graphs that are periodic or exhibit repetitive behavior. - Symmetry with respect to the pole:
The pole in polar coordinates corresponds to the origin in rectangular coordinates. To check for symmetry with respect to the pole, replace rrr with −r-r−r in the polar equation. If the equation is still valid after this change, the graph is symmetric with respect to the pole. This kind of symmetry is common in graphs like circles or certain types of rose curves, where a point on the graph can have a corresponding point at the opposite side of the pole.
These symmetry tests help to visualize the graph more easily without having to plot every point individually. However, it’s important to remember that an equation might fail one test but still exhibit that symmetry once the graph is completed over a full period (i.e., a complete rotation from 0 to 2π).
