Answer
Vertex: $(37,-6)$
Focus: $\left(36.75,-6\right)$
Directrix: $x=37.25$
See graph
Work Step by Step
We are given the parabola:
$y^2+12y=-x+1$
Put the equation in standard form:
$y^2+12y+36=-x+1+36$
$(y+6)^2=-(x-37)$
The standard equation is:
$(y-k)^2=4p(x-h)$
Determine $h,k,p$:
$h=37$
$k=-6$
$4p=-1\Rightarrow p=-0.25$
Determine the vertex:
$(h,k)=(37,-6)$
Determine the focus:
$(h+p,k)=\left(37+(-0.25),-6\right)=\left(36.75,-6\right)$
Determine the directrix:
$x=h-p$
$x=37-(-0.25)$
$x=37.25$
Determine the two points defining the latus rectum:
$x=36.75$
$(y+6)^2=-36.75+37$
$(y+6)^2=0.25$
$y+6=\pm 0.5$
$y+6=-0.5\Rightarrow y_1=-6.5$
$y+6=0.5\Rightarrow y_2=-5.5$
$\Rightarrow (36.75,-6.5),(36.75,-5.5)$
Plot the points, draw the directrix, and graph the parabola:
