Write the equation in standard form, given the following information.
vertex(-2,4), directrix y=1
well, you know that the equation for a parabola with vertex at (0,0) directrix y = -p is
x^2 = 4py
Your parabola has been shifted so the vertex is at (-2,4), so you want
(x+2)^2 = 4p(y-4)
Your directrix is at y = 1, which is 3 units below the vertex, so p=3
(x+2)^2 = 12(y-4)
see the graph at
http://www.wolframalpha.com/input/?i=parabola+(x%2B2)%5E2+%3D+12(y-4)
To determine the equation in standard form using the given information, we need to find the equation of the parabola with vertex (-2,4) and directrix y=1.
The standard form of a parabola equation is given by:
(y - k)^2 = 4p(x - h)
Where (h, k) represents the vertex, and p represents the distance between the vertex and either the focus or the directrix.
In this case, the vertex is (-2,4).
The given directrix is y=1, which means that the distance from the vertex to the directrix is 3 units (vertical distance from vertex to directrix).
Since the parabola is opening upwards, p is positive. So, p = 3.
Using this information, we can substitute the values into the standard form equation:
(y - 4)^2 = 4(3)(x - (-2))
Simplifying:
(y - 4)^2 = 12(x + 2)
Expanding the equation further:
y^2 - 8y + 16 = 12x + 24
Rearranging the equation in the standard form:
12x - y^2 + 8y = -8
Therefore, the equation of the parabola in standard form, with vertex (-2,4) and directrix y=1, is:
12x - y^2 + 8y = -8.
To write the equation of a parabola in standard form, we need the vertex and the directrix. The formula for a parabola in standard form is:
(y-k)^2 = 4a(x-h)
where (h,k) represents the coordinates of the vertex and "a" denotes the distance from the vertex to the focus and vertex to the directrix.
In this case, the vertex is (-2, 4) and the directrix is y=1. Since the directrix is a horizontal line, we can use the formula by considering the value of "a" as a positive number.
Step 1: Determine the value of "a":
The distance from the vertex (-2, 4) to the directrix y=1 is |4-1| = 3.
Step 2: Substitute the values into the standard form equation:
The equation becomes: (y-4)^2 = 4 * 3 * (x-(-2))
Simplifying further:
(y-4)^2 = 12(x+2)
Thus, the equation of the parabola in standard form is (y-4)^2 = 12(x+2).