Answer
See graph and explanations.
Work Step by Step
Step 1. We are given $y=\frac{x^4-1}{x^2}=x^2-\frac{1}{x^2}$, which can be considered as the sum of two functions.
Step 2. It is easier to graph the function by first plotting the known function $f(x)=x^2$, which is a parabola shown as the green curve in the figure and the function $g(x)=-\frac{1}{x^2}$, which is an exponential function symmetric around the y-axis shown as the purple curve in the figure.
Step 3. Find the first derivative: $y'=2x+\frac{2}{x^3}$ and $y'=0$ has no solution. The only critical point is when $x=0$ where $y'$ is undefined. We have the $y'$ sign changes as $..(-)..(0)..(+).$, which gives decreasing regions of $(-\infty,0)$ and increasing regions $(0,\infty)$.
Step 4. There is no asymptote for the function.
Step 5. To graph the given function from $f(x)=x^2$ and $g(x)=-\frac{1}{x^2}$, use the above information and add the corresponding y-coordinates at some key points and connect the points to form curves as shown in the figure (red curve).
