Final answer:
The sum of the x-coordinates of the intersection points of the graph f(x) = x2 with five lines parallel to the x-axis is determined by solving the equation for each y-value and summing these x-coordinates, accounting for the symmetry of the quadratic function across the y-axis.
Step-by-step explanation:
The student is asked about the intersection points of the graph of a quadratic function, f(x) = x2, with five lines that are parallel to the x-axis in the xy-plane. To find the sum of the x-coordinates of the points of intersection, one must solve the equation x2 = y for each y-value given by the lines. Since the lines are parallel to the x-axis, each will have a constant y-value. The quadratic function f(x) = x2 implies symmetry with respect to the y-axis. Thus, for each positive x that satisfies the equation, there is a corresponding negative x of the same absolute value. When these x-coordinates are summed, the positive and negative cancel out, leading to a sum of zero, provided that there is an equal amount of positive and negative intersections.
If the lines do not intersect the graph an equal number of times on both sides of the y-axis, then some x-coordinates will not have a corresponding pair to cancel out, and those values need to be summed to find the total sum of the x-coordinates of the points of intersection.