https://file.aiquickdraw.com/mathbot/file/1726209344762rcledyf1.jpg

The image contains several graphs, each representing a mathematical equation. The instructions ask to test for symmetry with respect to the x-axis, y-axis, and the origin. We are tasked with supporting the answer numerically (by visual inspection of the graph) and then confirming algebraically.

### Explanation of Symmetry:
1. **Symmetry with respect to the y-axis**:
   - If replacing \( x \) with \( -x \) results in the same equation, the graph is symmetric with respect to the y-axis.
   
2. **Symmetry with respect to the x-axis**:
   - If replacing \( y \) with \( -y \) results in the same equation, the graph is symmetric with respect to the x-axis.
   
3. **Symmetry with respect to the origin**:
   - If replacing \( x \) with \( -x \) and \( y \) with \( -y \) results in the same equation, the graph is symmetric with respect to the origin.

Let's go through the steps for a couple of the graphs to check for symmetry.

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### 24. \( x^2 + 4y^2 = 16 \) (Ellipse)
- **Graphical observation**:
   - This is an ellipse centered at the origin.
   - It is symmetric with respect to the x-axis, y-axis, and the origin.

- **Algebraic confirmation**:
   - Replacing \( x \) with \( -x \): \( (-x)^2 + 4y^2 = 16 \Rightarrow x^2 + 4y^2 = 16 \) (same equation, so y-axis symmetry).
   - Replacing \( y \) with \( -y \): \( x^2 + 4(-y)^2 = 16 \Rightarrow x^2 + 4y^2 = 16 \) (same equation, so x-axis symmetry).
   - Replacing both \( x \) with \( -x \) and \( y \) with \( -y \): Same result, confirming origin symmetry.

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### 25. \( y = \frac{1}{x} \) (Hyperbola)
- **Graphical observation**:
   - The graph is not symmetric with respect to the x-axis or y-axis, but it is symmetric with respect to the origin.

- **Algebraic confirmation**:
   - Replacing \( x \) with \( -x \): \( y = \frac{1}{-x} \) (not the same equation, so no y-axis symmetry).
   - Replacing \( y \) with \( -y \): \( -y = \frac{1}{x} \) (not the same equation, so no x-axis symmetry).
   - Replacing both \( x \) and \( y \): \( -y = \frac{1}{-x} \Rightarrow y = \frac{1}{x} \), confirming origin symmetry.

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Would you like me to continue analyzing all the graphs for symmetry, or would you prefer a detailed explanation for any specific ones? Let me know how you'd like to proceed.

### Additional Questions:
1. What is the symmetry of a graph of a circle equation like \( x^2 + y^2 = r^2 \)?
2. How can you identify rotational symmetry on a graph?
3. Why does the graph of \( y = x^3 \) have origin symmetry?
4. Can a function be symmetric with respect to both the x-axis and y-axis?
5. How does a graph's symmetry help in solving equations graphically?

### Tip:
For any function, symmetry can simplify your understanding and make graph sketching much easier. Symmetry reduces the need to calculate multiple points.

Use the graph of each equation to test for symmetry with respect to the x-axis, y-axis, and the origin. Support the answer numerically. Then confirm algebraically.

Analyze a variety of equations for symmetry with respect to the x-axis, y-axis, and the origin. Learn how to support your conclusions both numerically and algebraically with step-by-step explanations. Ideal for high school students studying geometry and algebra.

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