Without graphing, determine whether the following 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-x+2 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 \( y = x^2 - x + 2 \), 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**: For a graph to be symmetric about the x-axis, replacing \( y \) with \( -y \) should yield an equivalent equation. Let's test this:<br /><br /> \[<br /> y = x^2 - x + 2<br /> \]<br /><br /> Replace \( y \) with \( -y \):<br /><br /> \[<br /> -y = x^2 - x + 2<br /> \]<br /><br /> This is not equivalent to the original equation \( y = x^2 - x + 2 \). Therefore, the graph is **not** symmetric about the x-axis.<br /><br />2. **Symmetry with respect to the y-axis**: For a graph to be symmetric about the y-axis, replacing \( x \) with \( -x \) should yield an equivalent equation. Let's test this:<br /><br /> \[<br /> y = x^2 - x + 2<br /> \]<br /><br /> Replace \( x \) with \( -x \):<br /><br /> \[<br /> y = (-x)^2 - (-x) + 2 = x^2 + x + 2<br /> \]<br /><br /> This is not equivalent to the original equation \( y = x^2 - x + 2 \). Therefore, the graph is **not** symmetric about the y-axis.<br /><br />3. **Symmetry with respect to the origin**: For a graph to be symmetric about the origin, replacing \( x \) with \( -x \) and \( y \) with \( -y \) should yield an equivalent equation. Let's test this:<br /><br /> \[<br /> y = x^2 - x + 2<br /> \]<br /><br /> Replace \( x \) with \( -x \) and \( y \) with \( -y \):<br /><br /> \[<br /> -y = (-x)^2 - (-x) + 2 = x^2 + x + 2<br /> \]<br /><br /> This is not equivalent to the original equation \( y = x^2 - x + 2 \). Therefore, the graph is **not** symmetric about the origin.<br /><br />Since the graph is not symmetric with respect to the x-axis, y-axis, or the origin, the correct answer is:<br /><br />D. none of these