Parabolic Reflectors In flashlights, the reflector that directs the light from the bulb is parabolic. If the center of the bulb is placed at the focus of the reflector, then all the generated light will be directed forward in a concentrated beam. In a parabolic reflector, taking a cross section of the reflector through its central axis will create a parabola as shown in the diagram.

Part A
A manufacturer is designing a flashlight. For the flashlight to emit a focused beam, the bulb needs to be on the central axis of the parabolic reflector, 3 centimeters from the vertex. Write an equation that models the parabola formed when a cross section is taken through the reflector’s central axis. Assume that the vertex of the parabola is at the origin in the xy coordinate plane and the parabola can open in any direction.

Part B
Find the equation of the directrix of the parabola found in part A.

Part C
Based on feedback from an independent research firm, the flashlight manufacturer has decided to change the design of the flashlight. The reflector now needs to extend 4 centimeters past the center of the bulb, as shown in the diagram. In the new design, how wide will the reflector (CD) be at its widest point? Show your work.