Question

Determine whether the graph of the equation is symmetric with respect to the​ y-axis, the​ x-axis, the​ origin, more than one of​ these, or none of these.
y squared equals x squared plus 11
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Part 1
Select all that apply.
​x-axis
​y-axis
origin
none of these

Answer 1

To analyze the symmetry of the equation \( y^2 = x^2 + 11 \), we will test for symmetry with respect to the y-axis, the x-axis, and the origin.

1. **Symmetry with respect to the y-axis:**
To check for symmetry with respect to the y-axis, we replace \( x \) with \( -x \):

\[
y^2 = (-x)^2 + 11
\]
Simplifying, we get:
\[
y^2 = x^2 + 11
\]
This is the same equation as the original. Therefore, the graph is symmetric with respect to the y-axis.

2. **Symmetry with respect to the x-axis:**
To check for symmetry with respect to the x-axis, we replace \( y \) with \( -y \):

\[
(-y)^2 = x^2 + 11
\]
Simplifying, we get:
\[
y^2 = x^2 + 11
\]
This is also the same as the original equation. Therefore, the graph is symmetric with respect to the x-axis.

3. **Symmetry with respect to the origin:**
To check for symmetry with respect to the origin, we replace \( x \) with \( -x \) and \( y \) with \( -y \):

\[
(-y)^2 = (-x)^2 + 11
\]
Simplifying, we again get:
\[
y^2 = x^2 + 11
\]
This is the same as the original equation. Therefore, the graph is symmetric with respect to the origin.

Since the graph is symmetric with respect to the y-axis, the x-axis, and the origin, the correct selections are:

- x-axis
- y-axis
- origin

Thus, the graph is symmetric with respect to all three axes.