Question

Write the equation for the parabola. Vertex: (3,-1); Focus: (3,-4) An equation for this parabola is (Simplify your answer. Use integers or fractions for any numbers in the equation)

Answer 1

To find the equation of a parabola, we use the standard equation. In this case, the vertex is (3, -1) and the focus is (3, -4), so the equation is (x - 3)^2 = -12(y + 1).

Explanation:

To find the equation of a parabola with vertex (h, k) and focus (h, p), we can use the standard equation for a vertical parabola:

(x - h)^2 = 4a(y - k)

In this case, the vertex is (3, -1) and the focus is (3, -4). Since the x-coordinate of the vertex and the focus are the same, we know that the parabola opens vertically, either upwards or downwards. The vertex is at a higher y-value than the focus, so the parabola must open downwards.

Now, we need to find the value of a. We know that the distance between the vertex and the focus is the absolute value of a. The distance between (3, -1) and (3, -4) is 3, so |a| = 3. Since the parabola opens downwards, a = -3.

Now, we can plug the values of h, k, and a into the standard equation:

(x - 3)^2 = 4(-3)(y - (-1))

Simplify the equation:

(x - 3)^2 = -12(y + 1)