Question

What is the focus of the parabola x= -1/16 (y+3)^2 -6

(–10, –3)

(–3, –10)

(–3, –2)

(–2, 3)

Answer 1

\[
\begin{split}
x &=& -1/16 (y+3)^2 -6 \\
(x+6) &=& -1/16 (y+3)^2 \\
-16(x+6) &=& (y+3)^2
\end{split}
\]

Answer 2

http://www.mathwords.com/f/focus_parabola.htm

Answer 3

Can you read the Vertex with only a glance? You should be able to do that. Once we find the vertex, and the orientation of the parabola, the location of the Focus might be trivial!

Answer 4

The equation for a parabola with a focus at \((p, 0)\) is: \[
4px = y^2
\]

Answer 5

\[
\begin{split}
x &=& -1/16 (y+3)^2 -6 \\
(x+6) &=& -1/16 (y+3)^2 \\
-16(x+6) &=& (y+3)^2 \\
4(-4)(x+6) &=& (y+3)^2
\end{split}
\]

So as you can see, \(p=-4\).
But remember that our coordinates are shifted

Answer 6

or translated

Answer 7

oh, so how would that lead us to the foci?

Answer 8

or focus

Answer 9

You'd have to translate \((p,0)\) to based on how our equation was translated.

Answer 10

You need to get the equation is the form:

Vertical Parabola\[4p(y-k)=(x-h)^2\]Horizontal Parabola\[4p(x-h)=(y-k)^2\]Where (h,k) is the vertex and p is the distance from the vertex to the focus.

If the parabola opens up or down (vertical parabola) then the focus shares the x coordinate and you need to add p to the y coordinate accordingly.

If the parabola opens left or right (horizontal parabola) then the focus shares the y coordinate and you need to add p to the x coordinate accordingly

Answer 11

Immediately, x= -1/16 (y+3)^2 -6

The Vertex is (-6,-3)
-1/16 out front suggests it opens to the negative.

The Focus must look like this (x,-3)

These choices are eliminated.
(–3, –10)
(–3, –2)
(–2, 3)

Here's the winner.
(–10, –3)

Answer 12

Thanks everyone for your help :)