What is an equation of a parabola with the given vertex and focus?
vertex: (–2,5); focus: (–2, 6)
Since the vertex of the parabola is (-2,5) and the focus is (-2,6), the parabola opens upwards.
To find the equation of the parabola, we first need to find the equation of the directrix. Since the vertex is at (-2,5), the directrix will be the horizontal line that is one unit below the vertex.
Therefore, the directrix is y = 4.
The formula for the equation of a parabola with a vertical axis of symmetry and a focus (h, k+p) and directrix y = k-p is
(x-h)^2 = 4p(y-k).
Plugging in the values (h = -2, k = 5, p = 1) into the formula:
(x+2)^2 = 4(1)(y-5)
(x+2)^2 = 4(y-5)
(x+2)^2 = 4y - 20
x^2 + 4x + 4 = 4y - 20
4y = x^2 + 4x + 24
Therefore, the equation of the parabola is y = (1/4)x^2 + x + 6.