How do I Solve Parabolas?

Question

So I have to write the equation in standard form, and then state

(a) the vertex,

(b) the axis of symmetry, and

(c) the direction that the parabola open towards?

2x = y^2 + 10y + 32

How do I even begin to solve this? Please help. Thanks.

Answer 1

The y terms involve y^2, but the x terms do not involve x^2. That means the parabola has a horizontal axis.

The general equation for a parabola with a horizontal axis is:

(y - k)^2 = 4a(x - h) ...(1)

The vertex is (h, k).

The axis of symmetry is y = k.

If a is positive, the paraola opens right. If a is negative, the parabola opens left.

Complete the square on the y terms:

2x = y^2 + 10y + 32

2x - 32 = y^2 + 10y

2x - 32 + 25 = y^2 + 10y + 25

2x - 7 = (y + 5)^2

(y + 5)^2 = 2(x - 7 / 2).

Comparing this with (1):

h = 7 / 2, k = - 5, a = 1 / 2.

(a)

The vertex is (7 / 2, - 5).

(b)

The axis of symmetry is y = - 5.

(c)

a > 0. The parabola opens right.