\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. \\ \( \qquad \begin{array}{l}\begin{array}{ll}\text { (a) } x-y^{2}=5 \\ \text { Symmetry: } \\ \text { Symmetry: } \\ \square \text {-axis } \\ \square \text {-axis } \\ \square \text { none of the above }\end{array} \\ \square y \text {-axis } \\ \square \text { origin } \\ \square \text { none of the above }\end{array} \) \\ \hline\end{tabular}

To analyze the symmetries of the equation \( x - y^2 = 5 \), we can rearrange it to be in a form that makes checking symmetries easier: \( y^2 = x - 5 \). To check symmetry with respect to: 1. **The \( x \)-axis**: Replace \( y \) with \( -y \). We get \( x - (-y)^2 = 5 \), which simplifies to \( x - y^2 = 5 \). The equation remains unchanged, indicating symmetry about the \( x \)-axis. 2. **The \( y \)-axis**: Replace \( x \) with \( -x \). This gives us \( -x - y^2 = 5 \), or \( y^2 = -x - 5 \). This change does not yield the same equation, indicating no symmetry about the \( y \)-axis. 3. **The origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \). We get \( -x - (-y)^2 = 5 \), which simplifies to \( -x - y^2 = 5 \), or \( y^2 = -x - 5 \). This also does not yield the original equation, indicating no symmetry about the origin. Thus, the final answers for the symmetries are: - Symmetry with respect to the \( x \)-axis: Yes - Symmetry with respect to the \( y \)-axis: No - Symmetry with respect to the origin: No The results can be summarized as follows: - Symmetry with respect to the \( x \)-axis: ✔️ - Symmetry with respect to the \( y \)-axis: ❌ - Symmetry with respect to the origin: ❌