Question 692383
PART 1: The graph of the equation y^2-x^2+2x=2 is:
complete the square:
y^2-x^2+2x=2
y^2-(x^2-2x+1)=2-1
y^2-(x-1)^2=1
This is an equation of a hyperbola with vertical transverse axis and center at (1,0)
Its standard form: {{{(y-k)^2/a^2-(x-h)^2/b^2=1}}}, (h,k)=(x,y) coordinates of center
..
PART 2: The graph of the equation y-x^2=2x 
complete the square:
y-x^2=2x
y-x^2-2x=0
y-(x^2+2x+1)=0-1
y-(x+1)^2=-1
(x+1)^2=y+1
This is an equation of a parabola that opens upwards.
Its standard form: {{{(x-h)^2=4p(y-k)}}}, (h,k)=(x,y) coordinates of the vertex

note: Often the best way to see what conic the given equation is, write it in standard form by completing the square.