Find the standard form of the equation of the parabola with a focus at (0, -6) and a directrix at y = 6.

Question

imstuck · Answer

If the focus is at (0, -6), and the directrix is y = 6, what does that tell you about the vertex?  Where is the vertex situated in reference to the focus and the directrix?

anonymous · Answer

The vertex is 0,-6?

imstuck · Answer

The focus is (0, -6).  Do you know the "rule" about where the vertex sits in regards to where the focus and the directrix are?

anonymous · Answer

I have this chart that tells me that the vertex is (h,k). But the chart is the only thing I can remember at the moment.

imstuck · Answer

Ok, the vertex sits at the midpoint between the focus and the directrix.  If the focus is (0, -6), and the directrix is the line y = 6, then we need to find the midpoint of the line segment that both the focus and the directrix are on.  The coordinates of the focus are (0, -6), and the coordinate for the directrix, even though it is a line is (0, 6).  A straight horizontal line is a "y =" line with infinitely many x values, but we will use 0 as the x coordinate of the directrix.  This is because the focus has an x coordinate of 0 and they are on the same line.  Our line here is the y axis.  Is this making sense?

imstuck · Answer

|dw:1402189621053:dw|

anonymous · Answer

Yes. So would the mid point be something between 6 and -6? Like 0?

imstuck · Answer

Yes, right!  The parabola is centered at the origin!

imstuck · Answer

Do you know what type of parabola this is?  An x^2 one or a y^2 one?

anonymous · Answer

It's a x^2 because it's on the y-axis?

imstuck · Answer

Actually the parabolas that open like cups or curve down like a mountain are y^2 parabolas.  Just try to remember that, ok?  Is it a positive or a negative?  Look at my graph and tell me which it is according to where the directrix is and where the focus is.

imstuck · Answer

WAIT!!!!My fault!!! Youre right! It's an x^2, not a y^2

imstuck · Answer

Ok, sorry for that.  Typo, there.  Now, does it open up or down?

imstuck · Answer

Actually, look at this |dw:1402190152171:dw|The graph will ALWAYS "wrap itself around" the focus, away from the vertex.  In other words, the focus is always inside the parabola, while the directrix is always outside the parabola.  So which way does this open?  Up or down?  Look at what I drew.