how can i graph this function analytically? i can't properly use first derivative and second derivative helps me a bit but not enough

Question

$y={\frac{x^3-4}{(x-1)^2}}$, so I have to graph this equation analytically, but when I use the first derivative to find the maximum, or minimum, I get the very bad cubic equation. so I can't properly find the intervals, where the function is increasing or decreasing etc. but with second derivative I just find the point of inflection. so is there any good method to graph this equation.

Answer 1

Hint.

What you have is a rational function. The first step to sketch such functions is to find the vertical asymptotes and horizontal asymptotes, if any.

The general procedure is outlined here: https://tutorial.math.lamar.edu/classes/alg/graphrationalfcns.aspx

This function can be written as $$ f(x)=g(x)+\frac{3x-6}{(x-1)^2}=g(x)+\frac{3}{x-1}-\frac{3}{(x-1)^2} $$ where $g(x)=x+2$.

Observe that $$ \lim_{x\to +\infty}f(x)-g(x)=0,\quad \lim_{x\to -\infty}f(x)-g(x)=0 $$ and that $$ \lim_{x\to 1-}f(x)=\lim_{x\to 1+}f(x)=-\infty. $$