Determine the symmetries (if any) of the graph of the given relation. \( 7 y=2 x^{2}-3 \) Choose the correct symmetry of the graph. A. It is symmetric with respect to the origin. B. It is symmetric with respect to the \( x \)-axis. C. It is symmetric with respect to the \( y \)-axis. D. It is not symmetric with respect to the \( x \)-axis, \( y \)-axis, or origin.

To determine the symmetries of the graph of the relation \( 7y = 2x^2 - 3 \), let's analyze it step by step. First, let's rewrite the equation in a more familiar form: \[ y = \frac{2x^2 - 3}{7} \] This represents a parabola that opens upward with its vertex at \( (0, -\frac{3}{7}) \). **Checking for Symmetries:** 1. **Symmetry with respect to the \( y \)-axis:** - Replace \( x \) with \(-x\) in the equation: \[ y = \frac{2(-x)^2 - 3}{7} = \frac{2x^2 - 3}{7} = y \] - Since \( y(-x) = y(x) \), the graph is symmetric with respect to the \( y \)-axis. 2. **Symmetry with respect to the \( x \)-axis:** - Replace \( y \) with \(-y\) in the equation: \[ -y = \frac{2x^2 - 3}{7} \] - This does not simplify to the original equation unless \( y = 0 \), so the graph is **not** symmetric with respect to the \( x \)-axis. 3. **Symmetry with respect to the origin:** - Replace \( x \) with \(-x\) and \( y \) with \(-y\): \[ -y = \frac{2(-x)^2 - 3}{7} = \frac{2x^2 - 3}{7} \] - This does not result in \( -y = y \) (unless \( y = 0 \)), so the graph is **not** symmetric with respect to the origin. **Conclusion:** The graph of \( 7y = 2x^2 - 3 \) is **symmetric with respect to the \( y \)-axis**. **Answer:** C. It is symmetric with respect to the \( y \)-axis.