Write the equation of a parabola with a vertex at the origin and a focus at (4, 0).

find the general equation of the parabola with vertex at the origin and focus at (-4,0)

Sure, here's the equation of the parabola:

"Parabola: Y = X^2/16"

Remember, math may seem serious, but it's always good to put a little humor into it.

To write the equation of a parabola with a vertex at the origin and a focus at (4, 0), we need to determine the equation in terms of y.

The general equation of a parabola in standard form is:

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

where (h, k) represents the vertex, and p is the distance from the vertex to the focus or directrix line.

In this case, the vertex is at the origin (0, 0), and the focus is at (4, 0). To find p, we use the distance formula:

p = distance from vertex to focus
= √[(x₂ - x₁)^2 + (y₂ - y₁)^2]
= √[(4 - 0)^2 + (0 - 0)^2]
= √[16]
= 4

Now, we can substitute the values into the equation:

(x - 0)^2 = 4(4)(y - 0)

Simplifying, we get:

x^2 = 16y

Therefore, the equation of the parabola with a vertex at the origin and a focus at (4, 0) is x^2 = 16y.

To find the equation of a parabola with the vertex at the origin and the focus at (4, 0), we can use the standard form of a parabolic equation with the vertex at the origin, which is given by:

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

Where (h, k) represents the coordinates of the vertex and p is the distance from the vertex to the focus (also known as the focal length). In this case, h = 0 (since the vertex is at the origin) and k = 0 (since the vertex is also at the origin). Therefore, the equation simplifies to:

2. x^2 = 4p(y - 0)

Now, we need to find the value of p, which represents the distance from the vertex to the focus. In this case, the focus is at (4, 0), so the distance from the origin (vertex) to the focus is 4 units. Thus, p = 4.

Substituting the value of p into equation 2, we get:

3. x^2 = 4 * 4(y - 0)

Simplifying equation 3, we have:

4. x^2 = 16y

Therefore, the equation of the parabola with the vertex at the origin and focus at (4, 0) is x^2 = 16y.

V(0,0), F(4,0).

VF = 1/4a = 4-0 = 4
1/4a = 4
4a = 1/4
a = 1/16.

Vertex Form: X = a(y-k)^2 + h.
X = 1/16(y-0)^2 + 0
X = (1/16)y^2 = y^2/16.