https://file.aiquickdraw.com/mathbot/file/1729101758416u9gq26tt.jpg

Let's analyze the symmetries of each graph shown in the image:

### Graph 1 (The Circle):
- **x-axis symmetry**: Yes, the circle is symmetric about the x-axis because for every point \((x, y)\), the point \((x, -y)\) also exists.
- **y-axis symmetry**: Yes, the circle is symmetric about the y-axis because for every point \((x, y)\), the point \((-x, y)\) also exists.
- **Origin symmetry**: Yes, the circle is symmetric about the origin because for every point \((x, y)\), the point \((-x, -y)\) also exists.
- **Answer**: x-axis, y-axis, origin

### Graph 2 (The Parabola):
- **x-axis symmetry**: No, the parabola opens upwards, so it is not symmetric about the x-axis.
- **y-axis symmetry**: Yes, the parabola is symmetric about the y-axis because for every point \((x, y)\), the point \((-x, y)\) also exists.
- **Origin symmetry**: No, the parabola does not exhibit origin symmetry.
- **Answer**: y-axis

### Graph 3 (The Sine Wave):
- **x-axis symmetry**: No, the sine wave is not symmetric about the x-axis because it oscillates above and below the x-axis.
- **y-axis symmetry**: No, the sine wave is not symmetric about the y-axis because it is not mirrored over the y-axis.
- **Origin symmetry**: Yes, the sine wave is symmetric about the origin. For sine functions, \(\sin(-x) = -\sin(x)\), showing origin symmetry.
- **Answer**: Origin

Would you like more details on any of these graphs or symmetries?

Here are 5 related questions:
1. What is the mathematical condition for x-axis symmetry in a function?
2. How can you determine if a graph has y-axis symmetry just by looking at the equation?
3. Can a graph have both origin symmetry and y-axis symmetry at the same time?
4. How does the symmetry of a circle differ from that of an ellipse?
5. What are the implications of symmetry in solving equations graphically?

**Tip**: A graph has **origin symmetry** if flipping it across both the x- and y-axes results in the same shape. This happens when replacing \(x\) with \(-x\) and \(y\) with \(-y\) yields the same equation.

For each graph, select all symmetries that apply.

Learn how to analyze and determine the symmetries in various graphs, including a circle, parabola, and sine wave. This guide covers x-axis, y-axis, and origin symmetry, providing step-by-step explanations. Suitable for Grades 9-12.

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