Choose all of the equations that represent a parabola with the focus (0, 4) and the vertex (0, -6).

x^2 = 20(y+1)

None of the other equations represent a parabola with the given focus and vertex.

what is a focus and a vertex

In a parabola, the focus is the point on the interior of the parabola that is used to define its shape, while the vertex is the point at which the axis of symmetry intersects the parabola. The focus point is used to determine the distance from a point on the parabola to the focus and the distance from that point to the directrix line, which is important in defining the parabola's curve.

are you sure there are no other equation that match

Let's double-check to see if there are any other equations that represent a parabola with the focus (0, 4) and the vertex (0, -6):

The general form of the equation of a parabola with a vertical axis of symmetry is given by:

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

where (h,k) is the vertex and p is the distance between the focus and the vertex.

Given that the vertex is (0, -6), h = 0 and k = -6.

So, the equation becomes:

(x-0)^2 = 4p(y+6)
x^2 = 4p(y+6)

In this case, since the focus is (0, 4), it is 10 units away from the vertex. Therefore, p = 10.

Substitute p = 10 into the equation:
x^2 = 40(y+6)

Therefore, the equation x^2 = 40(y + 6) also represents a parabola with the focus (0, 4) and the vertex (0, -6).

So, the equations that represent a parabola with the focus (0, 4) and the vertex (0, -6) are:
1. x^2 = 20(y+1)
2. x^2 = 40(y+6)

1. x^2 = 20(y+1) has focus (0,4) and vertex (0,-1) so not that one

2. x^2 = 40(y+6) works ok

Apologies for the error in the initial response. Thank you for the correction.

The correct equation that represents a parabola with the focus (0, 4) and the vertex (0, -6) is:

2. x^2 = 40(y+6)

Thank you for pointing out the mistake.