Write an equation for the parabola !!!!!!!!!!!!!!!!!!!!!!1

Question

 with vertex (3, -6) and focus at (3, -4).

Answer 1

Given vertex (3 , -6)and focus (3 , -4) of a parabola

The focus is directly below the vertex, indicating a down-opening parabola

Vertex is (h , k) = (3 , -6)

Focus is (h , k-p) = (3 , -4)

==> k-p =-4

==> -6-p = -4

==> -p = -4+6

==> -p = 2

==> p = -2

==> 4p = 4(-2)

==> 4p = -8

Equation of parabola is

We know vertex is (3 , -6) and 4p = -8

Answer 2

Parabola vertex (3, -6) and focus at (3, -4).

Since the vertex and focus has same x - coordinate,

So the equation of parabola with vertex (h ,k ) is ( x - h )²  = 4p (y - k )

where p ≠ 0 .  The focus is at (h, k + p) .

The directrix is the line y = k - p .

The axis is the line x = h . If p > 0 , the parabola opens upward, and if p < 0 , the parabola opens downward.

(h , k ) = (3, -6)

k  = - 6 ---> (1)

(h , k + p ) = (3, -4)

k  + p  = - 4 ----> (2)

From (1) substitute the k value in equation (2).

- 6 + p  = - 4

p  = - 4 + 6

p  = 2

So p > 0 The parabola opens upward.

Substitute the values 4p  = 8 and (h , k ) = (3, - 6) in vertex form of vertical parabola.

( x - h )²  = 4p (y - k )

(x - 3)²  = 8(y - (-6))

(x - 3)² = 8(y + 6)

Parabola equation is (x - 3)² = 8(y + 6).