1. Write the equation of a parabola with a vertex at the origin and a focus at (4, 0).
a parabola with focus at (p,0) and vertex at (0,0) is given by
y^2 = 4px
Looks like y^2 = 16x
Well, I'm really bad at math, but I can certainly try! Let's see... how about we call this parabola "Parabeautiful"? It sounds so fancy!
So, the equation of a parabola with a vertex at the origin and a focus at (4, 0) can be written as:
y = (1/4)(x - 4)^2
And you know what? This Parabeautiful parabola is top-notch! It's so sleek and curvy, it might even outshine a clown's colorful wig!
To write the equation of a parabola with a vertex at the origin and a focus at (4, 0), we can use the standard form of the equation for a parabola:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex coordinates and p represents the distance from the vertex to the focus.
In this case, since the vertex is at the origin (h = 0, k = 0) and the focus is at (4, 0), we have:
(x - 0)^2 = 4p(y - 0)
which simplifies to:
x^2 = 4py
We can determine the value of p using the distance formula:
p = sqrt((x_f - x_v)^2 + (y_f - y_v)^2)
In this case, p = sqrt((4 - 0)^2 + (0 - 0)^2) = sqrt(16) = 4
Therefore, the equation of the parabola is:
x^2 = 4(4)y
or
x^2 = 16y
To write the equation of a parabola, we need to know its standard form. The standard form for a parabola with a vertex at the origin and a focus at (h, k) is given by:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex coordinates, and p represents the distance from the vertex to the focus.
In this case, since the vertex is at the origin (0, 0) and the focus is at (4, 0), we have:
h = 0
k = 0
p = 4
Substituting these values into the standard form equation, we get:
(x - 0)^2 = 4(4)(y - 0)
Simplifying:
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.