How do you write a vertex form equation?

Question

When the vertex is (10, 10) and the focus is (10, 12), how do you find and then write a vertex form equation of the parabola? 

Answer 1

Given a vertex at (10, 10) and a focus at (10, 12),

1) This is a parabola which opens upward, as you can see from the relationship between the points. The two x-values are the same. Therefore, the line of symmetry goes through this value (10).

1. The lowest point is the vertex at (10, 10).
2. The focus (inside the parabola) is above the vertex and located at (10, 12).

2) The y- distance between the vertex and focus is (12-10) =2, which gives a value p = 2.

3) Therefore, the parabola can be written as

(x-h)² = 4 * p * (y-k)

(x - 10)² = 4 * 2 * (y - 10)

(x - 10)² = 8 (y-10)

4) But we want to see the parabola in the vertex form, which is

y = (a - x)² + k

Therefore, we rewrite it like this, in order to algebraically isolate the variable y:

y - 10 = (1/8) (x - 10)²

y = (1/8) (x-10)² + 10

Try plugging this into a graphing calculator to compare the two forms of this equation.