sketch the graph y=x/(x-1)^2

Question

ketch the graph y=x/(x-1)^2

 

Not allowed to use graphing calculator. I am supposed to figure out how the graph is going to be by the "domain, intercepts, asymptotes, intervals of Increase/decrease, Local extrema and concavity and inflection points."

Answer 1

From Algebra II/College Algebra you see that degree of numerator is 1, degree of denominator is 2, therefore this rational algebraic function approaches 0 as x approaches ±infinity.  y = 0 (x-axis) is Horizontal Asymptote.

 

The denominator tells us that there is a Vertical Asymptote, x=1; 1 is not in the domain (every other Real value is).

y=0 when x=0...the Origin is on the graph.

 

Since the denominator, where defined, is always > 0, the y values are all negative if x<0 and all positive if x>0 (in domain).

 

function is increasing only on the interval (-1,1), based on y; = -(x+1)/(x-1)³

and there's an inflection point at x= -1.