Write an equation for a parabola with vertex at (3, -6) and focus at (3, -4)
See 5-2-12, 5:19pm post.
To write the equation for a parabola with a vertex and focus point, we need to understand the standard form of the equation.
A parabola equation in standard form is given by:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex of the parabola, and (h, k + p) is the focus point.
In this case, we are given that the vertex is (3, -6) and the focus is (3, -4). It means that the vertex point is (h, k) = (3, -6) and the focus point is (h, k + p) = (3, -4).
Using these values, we can determine the value of p.
From the given information, we know that the parabola opens vertically because the vertex point lies on the x-axis.
So, substituting the vertex coordinates into the standard form equation:
(x - h)^2 = 4p(y - k)
(x - 3)^2 = 4p(y - (-6))
(x - 3)^2 = 4p(y + 6)
Now we need to find the value of p. Remember that the focus point is (h, k + p) = (3, -4).
So, substituting these values into the equation, we get:
(3 - 3)^2 = 4p(-4 + 6)
0 = 4p(2)
0 = 8p
To find p, we divide both sides of the equation by 8:
0/8 = p
p = 0
Therefore, the value of p is 0.
Substituting p = 0 back into the equation, we have:
(x - 3)^2 = 4 * 0 * (y + 6)
(x - 3)^2 = 0
Simplifying further, we get:
(x - 3)^2 = 0
This equation represents a parabola with its vertex at (3, -6) and focus at (3, -4). Notice that the parabola is a horizontal line passing through the vertex, as p = 0 indicates that the parabola is degenerate.