Answer
See graph and explanations.
Work Step by Step
Step 1. Identify the domain of the function: rewrite the function as $y=\frac{x(x-1)+1}{x-1}=x+\frac{1}{x-1}$; we can identify the domain as $(-\infty,1)\cup (1,\infty)$.
Step 2. Take derivatives to get $y'=1-\frac{1}{(x-1)^2}$ and $y''=\frac{2}{(x-1)^3}$.
Step 3. We can find the possible critical points as $x=0,1,2$ (set $y'=0$). Check the signs of $y'$ across the critical points: $..(+)..(0)..(-)..(1)..(-)..(2)..(+)..$; thus the function increases on $(-\infty, 0),(2,\infty)$ and decreases on $(0,1),(1,2)$. A local maximum can be found at $y(0)=-1$. A local minimum can be found at $y(2)=3$.
Step 4. Check concavity across $x=1$ with signs of $y''$:$..(-)..(-1)..(+)..$; thus the function is concave down on $(-\infty,1)$ and concave up on $(1,\infty)$, but there is no inflection point as the function is not defined at $x=1$.
Step 5. We can identify a vertical asymptote as $x=1$, and a slant asymptote as $y=x$.
Step 6. The y-intercepts can be found by letting $x=0$ to get $y(0)=-1$.
Step 7. Based on the above results, we can graph the function as shown in the figure.
