Write an equation in standard form for each parabola.
Vertex (0, 0) and directrix y = 6.
y = -1/24x^2?
I agree.
verified here:
http://www.wolframalpha.com/input/?i=parabola+y%3D-1%2F24+x%5E2
To write the equation in standard form for a parabola with a given vertex and directrix, we can use the general equation for a parabola:
(y - k) = 4a(x - h)^2
where (h, k) is the vertex and a is the distance from the vertex to the focus and directrix.
Given that the vertex is (0, 0), we have h = 0 and k = 0.
The equation becomes:
(y - 0) = 4a(x - 0)^2
Simplifying further:
y = 4ax^2
Since the directrix is y = 6, which is 6 units above the vertex (0, 0), a = 6.
Substituting the value of a into the equation:
y = 4(6)x^2
Simplifying:
y = 24x^2
So, the equation in standard form for the given parabola is y = 24x^2.
To write the equation of a parabola in standard form given the vertex and the directrix, we can use the formula:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex, p represents the distance from the vertex to the focus (and also from the vertex to the directrix), and (x, y) represents any point on the parabola.
In this case, the vertex is (0, 0) and the directrix is y = 6.
Step 1: Find the value of p.
Since the directrix is a horizontal line, the distance from the vertex to the focus is the same as the distance from the vertex to the directrix. In this case, the distance is 6 units. Therefore, p = 6.
Step 2: Substitute the values into the formula and simplify.
(x - 0)^2 = 4(6)(y - 0)
x^2 = 24y
So, the equation of the parabola in standard form is x^2 = 24y.