Identify Critical Points
Key Questions

#(h,k)>(x,y)# represents the center of the hyperbola, ellipse, and circle.#(h,k)>(x,y)# represents the vertex of the parabola. 
If a hyperbola has an equation of the form
#{x^2}/{a^2}{y^2}/{b^2}=1# #(a>0, b>0)# , then its slant asymptotes are#y=pm b/ax# .Let us look at some details.
By observing,
#{x^2}/{a^2}{y^2}/{b^2}=1# by subtracting
#{x^2}/{a^2}# ,#Rightarrow {y^2}/{b^2}={x^2}/{a^2}+1# by multiplying by
#b^2# ,#Rightarrow y^2={b^2}/{a^2}x^2b^2# by taking the squareroot,
#Rightarrow y=pm sqrt{ {b^2}/{a^2}x^2b^2 } approx pm sqrt{{b^2}/{a^2}x^2}=pm b/a x# (Note that when
#x# is large,#b^2# is negligible.)Hence, its slant asymptotes are
#y=pm b/a x# . 
Critical points are points on the graph of a function where the first order derivative changes signs or equals to zero.
(Iam assuming you mean or you want something else )

Make the equation be in the form
#(x  h)^2/a^2  (y  k)^2/b^2 = 1# or
#(y  k)^2/a^2  (x  h)^2/b^2 = 1# If
#x# is on front, the hyperbola opens horizontally
If#y# is on front, the hyperbola opens vertically