Final answer:
The equation of a parabola to model the cross section of a flashlight reflector with the bulb at the focus 1/4 inch from the vertex and vertex at the origin is y = x^2.
Step-by-step explanation:
The question involves finding the equation of a parabola that models the cross section of a flashlight reflector, with its vertex at the origin and the bulb located at the focal point, which is 1/4 inch away from the vertex. This scenario is typical when using parabolic mirrors to create a concentrated beam of light, as in flashlights or car headlights.
The general equation for a parabola with a vertex at the origin and opening upward is y = ax^2. The focal length, f, is the distance from the vertex to the focus. Since the focus is 1/4 inch from the vertex, f = 1/4. We know that for a parabola, 4f = 1/a, or a = 1/(4f).
By substituting the value for f, we get a = 1/(4(1/4)) = 1.
Therefore, the equation of the parabola that models the reflector is y = x^2. This equation represents a parabola with its focus at (0, 1/4) that opens upward and reflects light rays emanating from the focus in a parallel beam outward, adhering to the reflective properties of a parabola.