Final answer:
To find the equation of the parabola, we need to determine the vertex, use the vertex form of the equation of a parabola, and solve for the value of a. The equation of the parabola with its focus at (-4, 7) and directrix y = 1 is y = -⅛(x + 4)² + 7 (option c).
Step-by-step explanation:
To find the equation of the parabola with its focus at (-4, 7) and directrix y = 1, we first need to determine the vertex of the parabola. The vertex of a parabola is the midpoint between its focus and directrix. In this case, the vertex is located at (-4, 4).
Next, we can use the vertex form of the equation of a parabola, which is given by y = a(x-h)² + k, where (h, k) is the vertex of the parabola. Plugging in the values of the vertex and the focus, we get the equation: y = a(x+4)² + 4.
Finally, we can solve for the value of a by substituting the coordinates of a point on the parabola. Since the point (0, 1) lies on the directrix, we can plug in these values and solve for a: 1 = a(0+4)² + 4. Solving for a, we get a = -1/8.
Therefore, the equation of the parabola with its focus at (-4, 7) and directrix y = 1 is y = -⅛(x + 4)² + 7 (option c).