Answer:
y = 1/8 x^2 - 5/4 x + 49/8.
Explanation:
The vertex is always halfway between the directrix and the focus and the parabola always curves away from the directrix so the equation will be of the form y = ax^2 + bx + c . It opens upwards. The absolute value of p will be the distance from the vertex to the focus and the distance between the vertex and the focus. So
|p| = 3-1 = 2. It is positive because the parabola opens upwards. The focus will be at (5, 3+2) = (5, 5).
Using the formula:
4p(y - k) = (x - h)^2 where (h, k) is the vertex, we have:
8(y - 3) = (x - 5)^2
Converting this to general form:
8y - 24 = x^2 - 10x + 25
8y = x^2 - 10x + 49
y = 1/8 x^2 - 5/4 x + 49/8