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The given equation is \( r = -3 + 5 \cos \theta \), which is in polar form. To determine the symmetry of the graph, we need to analyze it in polar coordinates:

1. **Symmetry about the x-axis** (polar symmetry about \(\theta = 0\)):
   - A polar graph is symmetric about the x-axis if replacing \(\theta\) with \(-\theta\) results in the same equation.
   - For this equation, replacing \(\theta\) with \(-\theta\):
     \[
     r = -3 + 5 \cos(-\theta) = -3 + 5 \cos(\theta)
     \]
     This shows that the equation remains unchanged, indicating symmetry about the x-axis.

2. **Symmetry about the y-axis** (polar symmetry about \(\theta = \frac{\pi}{2}\)):
   - To check for y-axis symmetry, replace \(r\) with \(-r\) and \(\theta\) with \(\pi - \theta\). Doing this results in:
     \[
     -r = -3 + 5 \cos(\pi - \theta) = -3 - 5 \cos(\theta)
     \]
     This changes the equation, indicating that the graph is **not symmetric about the y-axis**.

3. **Symmetry about the origin**:
   - Replace \(r\) with \(-r\) and check:
     \[
     -r = -3 + 5 \cos(\theta)
     \]
     This equation changes as well, showing no symmetry about the origin.

Therefore, the graph is symmetric **only about the x-axis**.

The correct answer is:
**b. x-axis only**

Would you like more details or further clarification?

Here are some related questions:
1. What other polar graphs exhibit symmetry about the x-axis?
2. How can you determine symmetry in polar graphs in general?
3. What happens to a graph when it is symmetric about the origin?
4. Can polar equations with sine functions exhibit y-axis symmetry?
5. How does the value of the constant term in the polar equation affect its symmetry?

**Tip**: Always check symmetry by testing transformations like \(\theta \rightarrow -\theta\) and \(r \rightarrow -r\) in polar equations.

The graph of r = -3 + 5cosθ is symmetric about the ________.

Analyze the symmetry of the polar equation r = -3 + 5cosθ. Learn how to determine symmetry about the x-axis, y-axis, and origin in polar graphs. This detailed explanation is suitable for high school students in Grades 10-12.

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