What are the vertex, focus, and directrix of the parabola with equation y = x^2 - 10x + 33?

Opens up so

(x-h) ^2=4P (y-k)

complete the square with the x term

take 1/2 of - 10 and square it and add that to both sides

y+25=x^2-10x+25+33

factor perfect square

y+25 = (x-5) ^2+33

minus 33 both sides

y-8 = (x-5) ^2

force factor out a 4

4 (1/4) (y-8) = (x-5) ^2

(x-5) ^2=4 (1/4) (y-8)

vertex is (5,8)

distance from directix is P or 1/4 or 0.25

since opens up, directix is ycoordinate-0.25 aka y=7.75 is directix

focus = (5,8+0.25) = (5,8.25)

vertex = (5,8)

directix: y=7.75

focus: (5,8.25)