Find the equation of a Parabola with Vertex: (-2,4) and directrix: y=7

x

7-4 = 3 and directrix is above vertex (sheds water) so a = -3

(x+2)^2 =-4*3 (y-4)^2 = -12(y-4)^2

=

Why did the parabola go to therapy?

Because it had some serious vertex issues! But don't worry, I'm here to help you out. The equation of a parabola with vertex (h,k) and directrix y=d is given by:

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

In this case, the vertex is (-2, 4) and the directrix is y=7. To find the value of "a", we need to calculate the distance between the vertex and the directrix.

The distance between a point (x, y) and a line with equation y = mx + c is given by:

Distance = |y - mx - c| / √(1 + m^2)

In this scenario, m = 0 (because the directrix is horizontal) and c = 7. Plugging in the values, we get:

Distance = |4 - 0(-2) - 7| / √(1 + 0^2)
= |4 + 7| / √(1)
= 11

Since the directrix is below the vertex, the value of "a" is negative. So, a = -11/4 in our equation.

Plugging in the vertex (-2, 4) and the value of "a" into the equation of the parabola, we get:

(y - 4)^2 = -11(x + 2)

And there you have it! The equation of the parabola is (y - 4)^2 = -11(x + 2). So, this parabola definitely has some "vertex" with therapy!

To find the equation of a parabola given its vertex and directrix, we can use the standard equation of a parabola with the vertex form: (x - h)^2 = 4p(y - k), where (h, k) represents the vertex coordinates, and p is the distance between the vertex and the focus/directrix.

In this case, the vertex is (-2, 4), and the directrix is y = 7. Since the directrix is above the vertex, the parabola opens downward.

Step 1: Determine the value of p. The distance between the vertex and the directrix is 3 units (7 - 4 = 3). As p is the distance between the vertex and focus/directrix, p = -3.

Step 2: Substitute the known values into the vertex form equation.

(x - (-2))^2 = 4(-3)(y - 4)

Simplifying,

(x + 2)^2 = -12(y - 4)

This is the equation of the parabola with the given vertex and directrix.

To find the equation of a parabola given its vertex and directrix, we can use the standard form of a parabolic equation.

The standard form of a parabolic equation with a vertex at point (h, k) and a vertical axis of symmetry is given by:

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

where p represents the distance between the vertex and the focus (p > 0 for a vertical parabola).

In this case, the given vertex is (-2, 4), so the equation becomes:

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

To find the value of p, we need to determine the distance between the vertex and the directrix, which is given as y = 7.

Since the directrix is a horizontal line, the distance between the vertex and the directrix is the difference in their y-coordinates. In this case:

Distance = Directrix - Vertex = 7 - 4 = 3

Therefore, the value of p is equal to half of this distance, so p = 3/2.

Substituting this value into the equation, we get:

(y - 4) = 4(3/2)(x + 2)^2
y - 4 = 6(x + 2)^2

Thus, the equation of the parabola with the given vertex (-2, 4) and directrix y = 7 is y - 4 = 6(x + 2)^2.