How to algebraically test polar graph symmetry for Pre-Calculus.

📚 Quick Study Guide

• 🧭 Symmetry with respect to the polar axis (x-axis): Replace $(r, \theta)$ with $(r, -\theta)$ or $(-r, \pi - \theta)$. If the equation remains unchanged, it has polar axis symmetry.
• 💫 Symmetry with respect to the line $\theta = \frac{\pi}{2}$ (y-axis): Replace $(r, \theta)$ with $(-r, -\theta)$ or $(r, \pi - \theta)$. If the equation remains unchanged, it has symmetry with respect to $\theta = \frac{\pi}{2}$.
• 🌌 Symmetry with respect to the pole (origin): Replace $(r, \theta)$ with $(-r, \theta)$ or $(r, \theta + \pi)$. If the equation remains unchanged, it has symmetry with respect to the pole.
• 🚨 Important Note: If a test fails, it *doesn't* necessarily mean there is no symmetry. It just means that particular test didn't confirm it. You might need to try other tests or graph the function.

Practice Quiz

1. Which test is used to check for symmetry with respect to the polar axis?

   1. Replace $(r, \theta)$ with $(r, \theta + \pi)$
   2. Replace $(r, \theta)$ with $(-r, \theta)$
   3. Replace $(r, \theta)$ with $(r, -\theta)$
   4. Replace $(r, \theta)$ with $(-r, \frac{\pi}{2} - \theta)$

2. Which substitution checks for symmetry with respect to the line $\theta = \frac{\pi}{2}$?

   1. $(r, \theta) \rightarrow (r, -\theta)$
   2. $(r, \theta) \rightarrow (-r, \theta + \pi)$
   3. $(r, \theta) \rightarrow (-r, -\theta)$
   4. $(r, \theta) \rightarrow (r, \theta - \pi)$

3. To test for symmetry with respect to the pole, you can replace $(r, \theta)$ with which of the following?

   1. $(r, -\theta)$
   2. $(-r, -\theta)$
   3. $(-r, \theta)$
   4. $(r, \frac{\pi}{2} - \theta)$

4. If replacing $(r, \theta)$ with $(r, -\theta)$ in the equation $r = 2 + 2\cos(\theta)$ yields the same equation, what type of symmetry does the graph have?

   1. Symmetry with respect to the pole
   2. Symmetry with respect to the line $\theta = \frac{\pi}{2}$
   3. Symmetry with respect to the polar axis
   4. No symmetry

5. Consider the polar equation $r^2 = \sin(2\theta)$. Which symmetry test is satisfied?

   1. Replacing $(r, \theta)$ with $(r, -\theta)$
   2. Replacing $(r, \theta)$ with $(-r, \theta)$
   3. Replacing $(r, \theta)$ with $(r, \pi - \theta)$
   4. None of the above

6. What can you conclude if the algebraic test for polar axis symmetry fails?

   1. The graph definitely does not have polar axis symmetry.
   2. The graph definitely has polar axis symmetry.
   3. The test is inconclusive; other tests or graphing are needed.
   4. The graph has symmetry with respect to the pole.

7. Which of the following polar equations is symmetric with respect to the line $\theta = \frac{\pi}{2}$?

   1. $r = 2\cos(\theta)$
   2. $r = 2\sin(\theta)$
   3. $r = \theta$
   4. $r = 2\cos(2\theta)$