jmc

Question

A parabola with equation y=ax^2+bx+c is reflected about the x-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of y=f(x) and y=g(x), respectively. Which of the following describes the graph of y=(f+g)(x)? (A) a parabola tangent to the x-axis (B) a parabola not tangent to the x-axis (C) a horizontal line (D) a non-horizontal line (E) the graph of a cubic function

Accepted Answer

Write the equation of the original parabola as y = a(x - h)^2 + k, where a 0. Then the equation of the reflected parabola is y = -a(x - h)^2 - k.When the parabolas are translated horizontally by 5 units, in opposite directions, their equations become y = a(x - h 5)^2 + k and y = -a(x - h 5)^2 - k.The sum of these expressions is 20ax 20ah = 20a (x - h),which is the equation of a non-horizontal line. The answer is (D).