Test the following equation for symmetry with respect to the \( x \)-axis, the \( y \)-axis, and the origin. \( y=x^{2}+5 \) Select all that apply. \( \square \) A. Symmetric with respect to the \( x \)-axis. \( \square \) B. Symmetric with respect to the \( y \)-axis. \( \square \) C. Symmetric with respect to the origin. \( \square \) D. None of these.

To test the equation \( y = x^2 + 5 \) for symmetry, we will check each type of symmetry one by one. **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 can obtain the original equation. Starting with: \[ -y = x^2 + 5 \] This simplifies to: \[ y = -x^2 - 5 \] This is not the same as the original equation \( y = x^2 + 5 \). Therefore, the equation is **not symmetric with respect to the \( x \)-axis**. **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 can obtain the original equation. Starting with: \[ y = (-x)^2 + 5 \] This simplifies to: \[ y = x^2 + 5 \] This is the same as the original equation. Therefore, the equation is **symmetric with respect to the \( y \)-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 \( -y \) in the equation and see if we can obtain the original equation. Starting with: \[ -y = (-x)^2 + 5 \] This simplifies to: \[ -y = x^2 + 5 \] or \[ y = -x^2 - 5 \] This is not the same as the original equation \( y = x^2 + 5 \). Therefore, the equation is **not symmetric with respect to the origin**. Based on the tests, we conclude: - The equation is **not symmetric with respect to the \( x \)-axis** (A is false). - The equation **is symmetric with respect to the \( y \)-axis** (B is true). - The equation is **not symmetric with respect to the origin** (C is false). Thus, the correct selection is: - \( \square \) A. Symmetric with respect to the \( x \)-axis. - \( \checked \) B. Symmetric with respect to the \( y \)-axis. - \( \square \) C. Symmetric with respect to the origin. - \( \square \) D. None of these.