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The image you uploaded is asking to test for symmetry in the given equations with respect to the x-axis, y-axis, and the origin. There are four equations listed. I'll go through each one to check for symmetry:

### Steps to Test Symmetry:
1. **Symmetry with respect to the x-axis**: Replace \( y \) with \( -y \). If the equation remains unchanged, it has x-axis symmetry.
2. **Symmetry with respect to the y-axis**: Replace \( x \) with \( -x \). If the equation remains unchanged, it has y-axis symmetry.
3. **Symmetry with respect to the origin**: Replace both \( x \) with \( -x \) and \( y \) with \( -y \). If the equation remains unchanged, it has origin symmetry.

### Let's analyze each equation:

1. **\( y = x^2 + 1 \)**
   - **X-axis**: Replace \( y \) with \( -y \): \( -y = x^2 + 1 \). This is not equivalent to the original equation, so no x-axis symmetry.
   - **Y-axis**: Replace \( x \) with \( -x \): \( y = (-x)^2 + 1 = x^2 + 1 \). The equation remains unchanged, so it has y-axis symmetry.
   - **Origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \): \( -y = (-x)^2 + 1 = x^2 + 1 \). This is not equivalent to the original equation, so no origin symmetry.

2. **\( y = x^3 - x \)**
   - **X-axis**: Replace \( y \) with \( -y \): \( -y = x^3 - x \). This is not equivalent to the original equation, so no x-axis symmetry.
   - **Y-axis**: Replace \( x \) with \( -x \): \( y = (-x)^3 - (-x) = -x^3 + x \). This is not equivalent to the original equation, so no y-axis symmetry.
   - **Origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \): \( -y = -x^3 + x \). This is equivalent to the equation obtained for the y-axis symmetry test, so it has origin symmetry.

3. **\( x = y^2 - 1 \)**
   - **X-axis**: Replace \( y \) with \( -y \): \( x = (-y)^2 - 1 = y^2 - 1 \). The equation remains unchanged, so it has x-axis symmetry.
   - **Y-axis**: Replace \( x \) with \( -x \): \( -x = y^2 - 1 \). This is not equivalent to the original equation, so no y-axis symmetry.
   - **Origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \): \( -x = (-y)^2 - 1 = y^2 - 1 \). This is equivalent to the equation obtained for the y-axis symmetry test, so no origin symmetry.

4. **\( y = 1 - x^2 \)**
   - **X-axis**: Replace \( y \) with \( -y \): \( -y = 1 - x^2 \). This is not equivalent to the original equation, so no x-axis symmetry.
   - **Y-axis**: Replace \( x \) with \( -x \): \( y = 1 - (-x)^2 = 1 - x^2 \). The equation remains unchanged, so it has y-axis symmetry.
   - **Origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \): \( -y = 1 - x^2 \). This is not equivalent to the original equation, so no origin symmetry.

### Summary of Symmetries:
1. \( y = x^2 + 1 \): **Y-axis Symmetry**
2. \( y = x^3 - x \): **Origin Symmetry**
3. \( x = y^2 - 1 \): **X-axis Symmetry**
4. \( y = 1 - x^2 \): **Y-axis Symmetry**

Would you like more details on any of these tests, or do you have any other questions?

### Related Questions:
1. What is the geometric interpretation of symmetry with respect to the x-axis, y-axis, and origin?
2. How does symmetry affect the graph of a function?
3. Can a function have more than one type of symmetry?
4. How do you determine symmetry in polar coordinates?
5. What are some common functions that exhibit symmetry, and what types?

### Tip:
Understanding symmetry can greatly simplify graphing functions, as symmetric graphs only require partial plotting to fully understand their shape.

Test for symmetry with respect to the x-axis, y-axis, and the origin for the following equations: 1) y = x^2 + 1, 2) y = x^3 - x, 3) x = y^2 - 1, 4) y = 1 - x^2.

Explore how to test equations for symmetry with respect to the x-axis, y-axis, and origin. Includes detailed analysis for equations like y = x^2 + 1, y = x^3 - x, x = y^2 - 1, and y = 1 - x^2. Suitable for high school students.

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