Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Complete a parabola and recover focus/directrix data
Problem
Complete the square for \(y~=~x^2~-~4x~+~7\). Give the vertex and canonical forms, vertex, \(p\)-value, axis, opening direction, focus, and directrix.
Big Picture
What this problem is really about
Complete the single squared-variable expression to expose vertex form, then rearrange into the canonical parabola form containing four times the signed focal distance. The squared variable tells the axis orientation, while the sign of that distance tells the opening direction. From the vertex, move the focal distance along the axis for the focus and the same distance oppositely for the directrix.
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