How do I graph these curves?

Question

My teacher taught me this in class but I still don't understand it.

Could someone please explain how to graph $y= x +\frac1x$ and $y = x - \frac1x$?

Thank you :)

Answer 1

I'm assuming you mean graphing by hand.

There are two terms right? That means you can just graph one term and then the other and then add their values after that (to make the final graph). So for the first example, $y=x+\frac{1}{x}$ you have $y=x$

and $y=\frac{1}{x}$

And now you add their contributions to "raising" the value of the total function:

The green curve is added of the two others. Note how the $\frac{1}{x}$-term dominates in the beginning, but when it nears $0$ for higher $x$, the $x$-term dominates instead, so the curve looks pretty much like a normal $y=x$-graph for high $x$.

Does it make sense? If not, let me know in the comments below.

Answer 2

Note that there're two asymptotes for both graphs. One is vertical $x=0$ which comes from the singularity of $\frac{1}{x}$. Another is an oblique $y=x$ which comes from the asymptotic behaviour as $x\to \infty$. In factor, two graphs are oblique conjugate hyperbolae.