how do i find the vertex, the focus, and the directrix of the parabola x^2-8x+4y+36=0

Perito

You put the equation in a "standard" form. This is taken from:

Parabola - Wikipedia, the free encyclopedia

"In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex (h,k), focus (h,k + p), and directrix y = k − p, with p being the distance from the vertex to the focus, has the equation with axis parallel to the y-axis



or, alternatively with axis parallel to the x-axis



More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form



such that , where all of the coefficients are real, where or , and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear equations"

So, for your equation,



The X is squared, so we'll try to get it into the first form





so h=4, p=-1, k=5

vertex (h,k) = (4,5)
focus (h,k + p) = (4, 4)
directrix y = k − p; y=6
and the axis is parallel to the y-axis