How do you write the equation of the parabola in vertex form given vertex (3,3) and focus: (-2,3)?
The vertex form for the equation of a parabola whose focus is shifted horizontally a signed distance,
Substitute the vertex
Compute the signed distance from the vertex to the focus:
Simplify the denominator:
First, let's figure out which way we need to draw the parabola.
The vertex is (3,3) and the focus is (-2,3).
These points have the same y-value, so they form a horizontal line. This means that the parabola will be horizontal.
It must be of the form
#x = a(y-k)^2+h#
Additionally, the focus is to the LEFT of the vertex, so the parabola will point to the LEFT (meaning
Let's call the distance between the focus and the vertex
We know that the value of
#a#is equal to #+-1/(4c)#.
In this case,
#c#is 5, since the vertex and focus are #5#units apart.
#a = +-1/(4(5)) = +- 1/20 = -1/20#since we already know #a#is negative.
This gives us everything we need to write our parabola's equation! The vertex
#x = a(y-k)^2 + h#
#x = -1/20(y-3)^2+3#