Final answer:
The equation of the parabola is y = (1/8)(x - 1)^2 + 2.
Step-by-step explanation:
The equation of a parabola in general form is y = ax^2 + bx + c. To find the equation of the parabola in this case, we need to determine the values of a, b, and c. Given that the directrix is x = -2, we can use the relationship between the directrix and the focus of a parabola to find the value of p, which is the distance between the directrix and the focus. Since the focus is at (4, 2), the distance from the directrix is 2 units. Therefore, p = 2. We also know that the vertex of the parabola lies halfway between the focus and the directrix, so the x-coordinate of the vertex is the average of -2 and 4, which is 1.
Using the formula for the equation of a parabola in terms of the vertex and focus, we have y = (1/4p)(x - h)^2 + k, where (h, k) is the vertex. Plugging in the values we found, we get y = (1/4*2)(x - 1)^2 + 2. Simplifying further, we have y = (1/8)(x - 1)^2 + 2. This is the equation of the parabola.