\begin{tabular}{l} For each equation, determine whether its graph is symmetric with respect to the $$ x $$-axis, the $$ y $$-axis, and the origin. \ Check all symmetries that apply. \ $$ \qquad \begin{array}{l}\begin{array}{ll} ext { (a) } x-y^{2}=5 \ ext { Symmetry: } \ ext { Symmetry: } \ \square ext {-axis } \ \square ext {-axis } \ \square ext { none of the above }\end{array} \ \square y ext {-axis } \ \square ext { origin } \ \square ext { none of the above }\end{array} $$ \ \hline\end{tabular}

Question

\begin{tabular}{l} For each equation, determine whether its graph is symmetric with respect to the $$ x $$-axis, the $$ y $$-axis, and the origin. \ Check all symmetries that apply. \ $$ \qquad \begin{array}{l}\begin{array}{ll}	ext { (a) } x-y^{2}=5 \ 	ext { Symmetry: } \ 	ext { Symmetry: } \ \square 	ext {-axis } \ \square 	ext {-axis } \ \square 	ext { none of the above }\end{array} \ \square y 	ext {-axis } \ \square 	ext { origin } \ \square 	ext { none of the above }\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To determine the symmetries of the equation $$ x - y^2 = 5 $$, we will check for symmetry with respect to the $$ x $$-axis, the $$ y $$-axis, and the origin.

### Step 1: Symmetry with respect to the $$ x $$-axis
To check for symmetry with respect to the $$ x $$-axis, we replace $$ y $$ with $$ -y $$ in the equation and see if we get the same equation.

1. Original equation: 
   $$
   x - y^2 = 5
   $$
2. Replace $$ y $$ with $$ -y $$:
   $$
   x - (-y)^2 = 5 \implies x - y^2 = 5
   $$

Since the equation remains unchanged, the graph is symmetric with respect to the $$ x $$-axis.

### Step 2: Symmetry with respect to the $$ y $$-axis
To check for symmetry with respect to the $$ y $$-axis, we replace $$ x $$ with $$ -x $$ in the equation and see if we get the same equation.

1. Original equation: 
   $$
   x - y^2 = 5
   $$
2. Replace $$ x $$ with $$ -x $$:
   $$
   -x - y^2 = 5 \implies -x - y^2 \neq 5
   $$

Since the equation changes, the graph is not symmetric with respect to the $$ y $$-axis.

### Step 3: Symmetry with respect to the origin
To check for symmetry with respect to the origin, we replace $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$ in the equation and see if we get the same equation.

1. Original equation: 
   $$
   x - y^2 = 5
   $$
2. Replace $$ x $$ with $$ -x $$ and $$ y $$ with $$ -y $$:
   $$
   -x - (-y)^2 = 5 \implies -x - y^2 = 5
   $$

Since the equation changes, the graph is not symmetric with respect to the origin.

### Conclusion
Based on the checks:
- Symmetry with respect to the $$ x $$-axis: **Yes**
- Symmetry with respect to the $$ y $$-axis: **No**
- Symmetry with respect to the origin: **No**

Thus, we can fill in the table as follows:

$$
\begin{array}{l}
\text{(a) } x - y^{2} = 5 \
\text{Symmetry: } \
\text{Symmetry: } \
\checkmark \text{ -axis } \
\square \text{ -axis } \
\square \text{ none of the above } \
\square y \text{ -axis } \
\square \text{ origin } \
\square \text{ none of the above }
\end{array}
$$
The graph of $$ x - y^2 = 5 $$ is symmetric with respect to the $$ x $$-axis but not with respect to the $$ y $$-axis or the origin.

For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin.

Check all symmetries that apply.

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For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin. Check all symmetries that apply.