Without graphing, determine whether the equation has a graph that is symmetric with respect to the x-axis, the y axis, the origin, or none of these. y=x^2+13 Select all that apply. A. x-axis B. y-axis C. origin D. none of these

To determine the symmetry of the graph of the equation \( y = x^2 + 13 \), we need to check for symmetry with respect to the x-axis, y-axis, and the origin.<br /><br />1. **Symmetry with respect to the x-axis:**<br /> - To test for symmetry with respect to the x-axis, we replace \( y \) with \( -y \) and see if the equation remains unchanged.<br /> - Original equation: \( y = x^2 + 13 \)<br /> - Replace \( y \) with \( -y \): \( -y = x^2 + 13 \)<br /> - This is not the same as the original equation, so the graph is not symmetric with respect to the x-axis.<br /><br />2. **Symmetry with respect to the y-axis:**<br /> - To test for symmetry with respect to the y-axis, we replace \( x \) with \( -x \) and see if the equation remains unchanged.<br /> - Original equation: \( y = x^2 + 13 \)<br /> - Replace \( x \) with \( -x \): \( y = (-x)^2 + 13 \)<br /> - Since \( (-x)^2 = x^2 \), the equation remains unchanged: \( y = x^2 + 13 \)<br /> - Therefore, the graph is symmetric with respect to the y-axis.<br /><br />3. **Symmetry with respect to the origin:**<br /> - To test for symmetry with respect to the origin, we replace \( x \) with \( -x \) and \( y \) with \( -y \) and see if the equation remains unchanged.<br /> - Original equation: \( y = x^2 + 13 \)<br /> - Replace \( x \) with \( -x \) and \( y \) with \( -y \): \( -y = (-x)^2 + 13 \)<br /> - Since \( (-x)^2 = x^2 \), the equation becomes: \( -y = x^2 + 13 \)<br /> - This is not the same as the original equation, so the graph is not symmetric with respect to the origin.<br /><br />Based on this analysis, the correct answer is:<br /><br />B. y-axis