Determine if the given relation \( x=1 \cdot y^{2}+4 \) is symmetrical to the \( x \)-axis, \( y \)-axis, the origin, or none. origin symmetry \( x \)-axis symmetry \( y \)-axis symmetry none

To determine the symmetry of the relation \( x = y^{2} + 4 \), let's analyze it with respect to the three types of symmetries: \( y \)-axis, \( x \)-axis, and the origin. 1. **\( y \)-Axis Symmetry:** - **Test:** Replace \( x \) with \( -x \) in the equation. - **Result:** \( -x = y^{2} + 4 \) - **Conclusion:** This is not equivalent to the original equation \( x = y^{2} + 4 \). Therefore, the relation **does not** have \( y \)-axis symmetry. 2. **\( x \)-Axis Symmetry:** - **Test:** Replace \( y \) with \( -y \) in the equation. - **Result:** \( x = (-y)^{2} + 4 \) simplifies to \( x = y^{2} + 4 \) - **Conclusion:** This is identical to the original equation. Therefore, the relation **does** have \( x \)-axis symmetry. 3. **Origin Symmetry:** - **Test:** Replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation. - **Result:** \( -x = (-y)^{2} + 4 \) simplifies to \( -x = y^{2} + 4 \) - **Conclusion:** This is not equivalent to the original equation \( x = y^{2} + 4 \). Therefore, the relation **does not** have origin symmetry. **Final Answer:** The relation is symmetric about the \( x \)-axis.