Identify Critical Points
Key Questions
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#(h,k)->(x,y)# represents the center of the hyperbola, ellipse, and circle.#(h,k)->(x,y)# represents the vertex of the parabola.
AJ Speller
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Sep 28 2014
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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^2-b^2# by taking the square-root,
#Rightarrow y=pm sqrt{ {b^2}/{a^2}x^2-b^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# .
Wataru
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Sep 28 2014
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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 )
Konstantinos Michailidis
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1
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Sep 20 2015
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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
seph
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5
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Oct 12 2014