when writing an equation in slope-intercept form containg H(2,2), parallel to the graph of y=x-3, does it become y-2=x^2-2x
no.
Write the equation in slope intercept form
y=mx+b
you know the slope is 1 (from the given equation).
y=x+b but it contains the point 2,2
2=2+b and b then is zero.
the equation is ...
y=x