Answer
See graph and explanations.
Work Step by Step
Step 1. Given $y=\frac{x^2}{x^2-4}=1+\frac{4}{x^2-4}$, we have $y'=-\frac{2x}{(x^2-4)^2}$
Step 2. Letting $y'=0$, we have $x=0$; the function is undefined at $x=\pm2$, so the critical points are $x=0, \pm2$
Step 3. We have the $y'$ sign changes as $..(+)..(-2)..(+)..(0)..(-)..(2)..(-)..$ which gives increasing regions of $(-\infty,-2), (-2,0)$ and decreasing regions $(0,2),(2,\infty)$. There is a local maximum at $(0,0)$.
Step 4.The function has three asymptotes: $x=\pm2, y=1$
Step 5. Graph the function based on the above information as shown in the figure (red curve).
