Find an equation of a parabola with a vertex at the origin and directrix

y= -4

Recall that the parabola

x^2 = 4py

has its vertex at (0,0) and its directrix at y = -p.

Well, let's start with the standard equation for a parabola with a vertical axis of symmetry: y = ax^2 + bx + c. Since the vertex is at the origin (0, 0), we know that c = 0.

Now, the distance between the vertex and the directrix is the same as the distance between the vertex and any point on the parabola. In this case, the distance from the origin to the directrix y = -4 is 4 units.

To find the value of a, we can use the formula: a = 1 / (4p), where p is the distance from the vertex to the focus. Since the focus is also 4 units above the vertex, p = 4. Therefore, a = 1 / (4 * 4) = 1/16.

Now we have our equation: y = (1/16)x^2 + bx

To find the value of b, we can use the fact that the directrix is a horizontal line, so the focus is also 4 units above the directrix. This means the focus has coordinates (0, 4).

Using the formula for the focus of a parabola, (h, k + p), we can plug in the values and solve for b:

(0, 4) = (h, k + p)
(0, 4) = (0, 0 + 4)

This tells us that b = 0.

Finally, we have our equation: y = (1/16)x^2

And that's it! The equation of the parabola with a vertex at the origin and directrix y = -4 is y = (1/16)x^2.

To find the equation of a parabola with a vertex at the origin (0,0) and a given directrix y = -4, we can use the standard form of a parabola equation.

The standard form of the equation for a parabola with a vertex at (h, k) is:

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

In this case, since the vertex is at the origin (0,0), the equation becomes:

y = ax^2

To find the value of "a", we need to consider the distance between the vertex and the directrix. The distance between the vertex and the directrix is equal to the distance between the vertex and the focus. In this case, the vertex is at (0,0) and the directrix is y = -4, so the distance is 4 units.

Since the parabola opens upwards (because the coefficient of x^2 is positive), the focus lies above the vertex. The focus will have the same x-coordinate as the vertex but a different y-coordinate. Let's call the coordinates of the focus (0, f).

Using the distance formula, we know that the distance between the vertex (0,0) and the focus (0,f) is given by:

sqrt((0 - 0)^2 + (f - 0)^2) = 4

Simplifying the equation, we have:

sqrt(f^2) = 4

f = ±4

We take the positive value since the focus lies above the vertex, so f = 4.

Therefore, the equation of the parabola with a vertex at the origin and directrix y = -4 is:

y = 4x^2

To find an equation of a parabola with a vertex at the origin and a directrix y = -4, we can use the definition of a parabola and the distance formula.

The definition of a parabola states that for any point on the parabola, the distance from the point to the focus is equal to the distance from the point to the directrix.

Considering the vertex at the origin (0,0), we can determine the focus to be located at the point (0, d) where d is the distance between the vertex and the directrix. Since the directrix is y = -4, the distance d is 4.

Now, let's denote any point on the parabola as (x, y). According to the definition, we have:

√(x^2 + y^2) = √(x^2 + (y - (-4))^2)

Squaring both sides of the equation, we get:

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

Expanding the equation, we have:

y^2 = (y + 4)^2

Using the FOIL method to multiply the right side, we have:

y^2 = y^2 + 8y + 16

Simplifying the equation, we get:

0 = 8y + 16

Subtracting 16 from both sides, we have:

-16 = 8y

Dividing both sides by 8, we get:

-2 = y

Therefore, the equation of the parabola with a vertex at the origin and a directrix y = -4 is:

y = -2