Question

when testing a polar equation for symmetry, there are three types of symmetry that you need to test for. when testing each of these types, you must substitute some values of the equation to see if you get an equivalent equation. please match up each type of symmetry with the values you need to replace in the equation. for symmetry about the polar axis, choose... for symmetry about the line ( \theta = \frac{pi}{2} ), choose... for symmetry about the pole, replace r with -r and θ with -θ replace r with -r replace θ with -θ

Explanation:

• Polar Axis Symmetry: To test if a polar curve is symmetric about the polar axis (the x - axis in Cartesian terms), we substitute $\theta$ with $-\theta$. This is because the polar axis is like the x - axis, and reflecting over it changes the angle $\theta$ to $-\theta$ while the radius $r$ (or $-r$) should still satisfy the equation if symmetric.
• Line $\boldsymbol{\theta=\frac{\pi}{2}}$ Symmetry: The line $\theta = \frac{\pi}{2}$ is the y - axis in Cartesian terms. One way to test symmetry about this line is to substitute $r$ with $-r$ and $\theta$ with $-\theta$. Geometrically, this transformation reflects the point $(r,\theta)$ across the line $\theta=\frac{\pi}{2}$.
• Pole Symmetry: The pole is the origin in polar coordinates. To test symmetry about the pole, we substitute $r$ with $-r$. If replacing $r$ with $-r$ gives an equivalent equation, the curve is symmetric about the pole, as the pole is the center and changing the sign of $r$ (while keeping $\theta$ the same or adjusting appropriately) should still lie on the curve if symmetric about the pole.

Answer:

• For symmetry about the polar axis: replace $\theta$ with $-\theta$
• For symmetry about the line $\theta=\frac{\pi}{2}$: replace $r$ with $-r$ and $\theta$ with $-\theta$ (or equivalently, replace $\theta$ with $\pi - \theta$; but among the given options, "replace $r$ with $-r$ and $\theta$ with $-\theta$" is the match)
• For symmetry about the pole: replace $r$ with $-r$