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.