Question

What is the focus of the parabola? y=−1/4x2−x+3

Answer 1

Answer:  Focus = (-2, 3)

Step-by-step explanation:

$y=-\dfrac{1}{4}x^2-x+3\\\\\rightarrow a=-\dfrac{1}{4},\ b=-1$

First let's find the vertex. We do that by finding the Axis-Of-Symmetry:

$AOS: x=\dfrac{-b}{2a}\quad =\dfrac{-(-1)}{2(\frac{-1}{4})}=\dfrac{1}{-\frac{1}{2}}=-2$

Then finding the maximum by inputting x = -2 into the given equation:

$y=-\dfrac{1}{4}(-2)^2-(-2)+3\\\\y=-1+2+3\\\\y=4$

The vertex is: (-2, 4)

Now let's find p, which is the distance from the vertex to the focus:

$a=\dfrac{1}{4p}\\\\\\-\dfrac{1}{4}=\dfrac{1}{4p}\\\\\\p=-1$

The vertex is (-2, 4) and p = -1

The focus is (-2, 4 + p) = (-2, 4 - 1) = (-2, 3)