Answer
See graph and explanations.
Work Step by Step
Step 1. Identify the domain of the function: rewriting 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 decreases on $(-\infty, 0),(2,\infty)$ and increases on $(0,1),(1,2)$. A local minimum can be found at $y(0)=1$. A local maximum can be found at $y(2)=-3$.
Step 4. Check concavity across $x=1$ with signs of $y''$:$..(+)..(-1)..(-)..$; thus the function is concave up on $(-\infty,1)$ and concave down 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.
