In Exercises 7–12, test for symmetry with respect to a. the polar axis. b. the line θ=π2. c. the pole. r = 4 + 3 cos θ

Verified step by step guidance

1. **Step 1:** To test for symmetry with respect to the polar axis, replace $\theta$ with $-\theta$ in the equation $r=4+3\cos \theta$. Simplify the expression to see if it results in the original equation.
2. **Step 2:** To test for symmetry with respect to the line $\theta =\frac{\pi }{2}$, replace $\theta$ with $\pi -\theta$ in the equation $r=4+3\cos \theta$. Simplify the expression to check if it results in the original equation.
3. **Step 3:** To test for symmetry with respect to the pole, replace $r$ with $-r$ in the equation $r=4+3\cos \theta$. Simplify the expression to see if it results in the original equation.
4. **Step 4:** Analyze the results from each symmetry test. If the modified equation matches the original equation in any of the tests, the graph is symmetric with respect to that axis, line, or point.
5. **Step 5:** Summarize the findings for each symmetry test, indicating whether the graph is symmetric with respect to the polar axis, the line $\theta =\frac{\pi }{2}$, or the pole.

Key Concepts

Symmetry in Polar Coordinates

In polar coordinates, symmetry can be analyzed with respect to the polar axis, the line θ=π/2, and the pole. A graph is symmetric about the polar axis if replacing θ with -θ yields the same equation. Symmetry about the line θ=π/2 occurs if replacing θ with π-θ results in the same equation, while symmetry about the pole is confirmed if replacing r with -r and θ with θ+π gives the same equation.

Polar Axis

The polar axis is the horizontal line in polar coordinates, equivalent to the positive x-axis in Cartesian coordinates. To test for symmetry with respect to the polar axis, one substitutes -θ into the polar equation. If the resulting equation is equivalent to the original, the graph exhibits symmetry about the polar axis, indicating that it is a mirror image across this line.

The Pole

The pole in polar coordinates is the origin point, represented by r=0. To check for symmetry about the pole, one replaces r with -r and θ with θ+π in the polar equation. If the modified equation remains unchanged, the graph is symmetric about the pole, suggesting that points on the graph have corresponding points directly opposite them through the origin.