Identify the vertex, axis of symmetry, \( y \)-intercept, \( x \)-intercepts, and opening of the parabola, then sketch the graph. \( y=x^{2}-x \)

We’re given the quadratic function   y = x² – x. Below are the steps to analyze its features: 1. • Vertex:   – Write the quadratic in the form y = ax² + bx + c where a = 1, b = –1, and c = 0.   – The x-coordinate of the vertex is found by     h = –b/(2a) = –(–1)/(2·1) = 1/2.   – Substitute x = 1/2 back into the equation to find the y-coordinate:     y = (1/2)² – (1/2) = 1/4 – 1/2 = –1/4.   – Thus, the vertex is at (1/2, –1/4). 2. • Axis of Symmetry:   – The axis of symmetry is the vertical line that passes through the vertex.   – It is given by x = 1/2. 3. • y-intercept:   – The y-intercept occurs when x = 0.   – Substitute x = 0 into the equation: y = 0² – 0 = 0.   – So the y-intercept is at (0, 0). 4. • x-intercepts:   – The x-intercepts occur when y = 0.   – Set up the equation: 0 = x² – x.   – Factor: 0 = x(x – 1).   – Solve: x = 0 or x – 1 = 0 → x = 1.   – Therefore, the x-intercepts are at (0, 0) and (1, 0). 5. • Opening of the Parabola:   – Since the coefficient a = 1 is positive, the parabola opens upward. 6. • Sketching the Graph:   – Plot the vertex at (1/2, –1/4).   – Draw the axis of symmetry as a dashed vertical line through x = 1/2.   – Plot the intercepts: the y-intercept and one of the x-intercepts are both at (0, 0) while the other x-intercept is at (1, 0).   – Notice that (0, 0) is on the parabola and by symmetry the point (1, 0) is its mirror image about the line x = 1/2.   – For additional points, choose values equidistant from the axis. For example:     • When x = –0.5 (which is 1 unit left of x = 0.5),       y = (–0.5)² – (–0.5) = 0.25 + 0.5 = 0.75.     • Its symmetric point is x = 1.5 (1 unit right of x = 0.5), yielding:       y = (1.5)² – 1.5 = 2.25 – 1.5 = 0.75.   – Plot these points and sketch the smooth, curved U-shaped graph opening upward. Summary of key features:   • Vertex: (1/2, –1/4)   • Axis of symmetry: x = 1/2   • y-intercept: (0, 0)   • x-intercepts: (0, 0) and (1, 0)   • Opening: Upward This analysis provides all the necessary details to sketch an accurate graph of the parabola y = x² – x.