Final answer:
To write the equation in vertex form for the parabola with vertex (0, -5) and directrix y = -3, use the formula x = h - (1 / (4a)) (y - k)^2, where (h, k) is the vertex and a is the coefficient of the x^2 term. Rearrange the equation to y = (4a)x^2 - 20ax + 15a - 5.
Step-by-step explanation:
To write the equation in vertex form for the parabola with vertex (0, –5) and directrix y = –3, we can use the formula:
x = h - (1 / (4a)) (y - k)^2
where (h, k) is the vertex of the parabola and a is the coefficient of the x^2 term. In this case, the vertex is (0, –5), so h = 0 and k = –5. The directrix is y = –3, which means the parabola opens downwards. Therefore, a is negative.
Plugging in the values into the formula, we get:
x = 0 - (1 / (4a)) (y - (–5))^2
Since the vertex form of a parabola is y = a(x – h)^2 + k, we can rearrange the equation to:
y = (4a)x^2 - 20ax + 15a - 5