Prove that the graph of the equation is a​ parabola, and find its​ vertex, focus, and directrix. x^2+2x−y+2=0

Question

Prove that the graph of the equation is a​ parabola, and find its​ vertex, focus, and directrix.

x^2+2x−y+2=0

Answer 1

y=x^2 +2x +2

an upward opening parabola

use a graphing calculator, handheld or online

= x^2 +2x +1 +2 -1

= (x+1)^2 + 1 with vertex = (-1, 1)

axis of symmetry x=-1

with y intercept (0,2)

x intercepts DNE, Do Not Exist, as the parabola is always above the x axis

roots are imaginary = -1+/-i

focus = (-1, 5/4)

directrix y = 3/4

both above the x axis

vertex is half way between the focus and directrix= (-1,1)

5/4 + 3/4 = 8/4 =2 divided by 2 = 1 = y coordinate of the vertex