8. Raj bought a new super bright flashlight for his camping trip. Use your knowledge of circles and parabolas to define equations that model the beam of light and the reflector within the flashlight.
Part I: Raj tested his new flashlight by shining it on his bedroom wall. The beam of light can be described by the equation . How many inches wide is the beam of light on the wall?
Step 1: How many inches wide is the beam of light on the wall? Explain your reasoning. (4 points)
Step 2: Graph the beam of light on the wall on the coordinate grid below. (4 points)
Part II: Flashlights have parabolic reflectors that allow the reflected light from the bulb’s filament (focus) to be directed out in parallel beams of light. If the filament of Raj’s flashlight is located at (3, 4) and the directrix is at x = 1, write an equation for the cross section of his flashlight.
Step 1: If the focus is (3, 4) and the directrix is x = 1, identify the location of the vertex. Explain your thinking. (3 points)
Step 2: The focus, directrix, and vertex can be defined in terms of the variables h, p, and k. Use the information from Step 1 to define h, p, and k for the equation that models the cross section of Raj’s flashlight. Show your work. (2 points)
Focus (h + p, k)
Directrix x = h – p
Vertex (h, k)
Step 3: Horizontal parabolas can be modeled by the equation . Use the information from Step 2 to write the equation for the cross section of Raj’s flashlight. Simplify your answer by writing the equation in standard form, . Show your work. (4 points)