How do I find the focus of the parabola x-3=-2((y+4)^2)+3?

Question

imstuck · Answer

First let's figure out this parabola on a graph.  We have a +3 on the right side that we need to move over to the other side with the x's to get this into standard form.  That gives us x + 6, which is one coordinate of the vertex.  The other is inside the parenthesis with the y.  The vertex of the parabola sits at (-6, -4).  this is a y^2 parabola, and it is negative, so it lies at (-6, -4) running horizontal to the x axis.  Like this|dw:1401570297307:dw|To find the focus, use the standard form for a parabola that runs horizontal to the x axis$$(y-k)^{2}=4p(x-h)$$The "4p" part is very important to use in finding the focus.  We use this fact and fill in what we know with the fact that -2 is the coordinate in front of the x term.  The equation would be 4p = -2 and solve for p.  P = -1/2 and that is the DISTANCE from the vertex that the focal point is on the x axis so add the point to the corresponding coordinate.  This focal point lies on the x axis, remember, so it gets added to the x coordinate of the center.  If the center is at (-6, -4), the focal point's coordinates would be (-6.5, -4).  I know that was really confusing, but these ARE quite confusing!  If you have other questions, or need more help, just post and we're here to help you!

dan815 · Answer

focus is hte special point where all lines refeclected on the side of the parabola pass through

dan815 · Answer

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