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what are the vertex, focus, and directrix of the parabola with equation y=x^2-8x+18?
The "general" form of a parabola's equation is the one you're used to, y = ax2 + bx + c — unless the quadratic is "sideways", in which case the equation will look something like x = ay2 + by + c. The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the x part is squared; for sideways (horizontal) parabolas, the y part is squared. The "vertex" form of a parabola with its vertex at (h, k) is: regular: y = a(x – h)2 + k sideways: x = a(y – k)2 + h Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved The conics form of the parabola equation (the one you'll find in advanced or older texts) is: regular: 4p(y – k) = (x – h)2 sideways: 4p(x – h) = (y – k)2 Why "(h, k)" for the vertex? Why "p" instead of "a" in the old-time conics formula? Dunno. The important thing to notice, though, is that the h always stays with the x, that the k always stays with the y, and that the p is always on the unsquared variable part. The relationship between the "vertex" form of the equation and the "conics" form of the equation is nothing more than a rearrangement: y = a(x – h)2 + k y – k = a(x – h)2 (1/a)(y – k) = (x – h)2 4p(y – k) = (x – h)2 In other words, the value of 4p is actually the same as the value of 1/a; they're just two ways of saying the exact same thing. But this new variable p is one you'll need to be able to work with when you're doing parabolas in the context of conics: it represents the distance between the vertex and the focus, and also the same (that is, equal) distance between the vertex and the directrix. And 2p is then clearly the distance between the focus and the directrix. I obtained this from: http://www.purplemath.com/modules/parabola.htm
what is an equation in standard form of an ellipse centered at the origin with vertex (-7,0) and co-vertex (0,5)?