How do you plot the graph of the function $y^2 = \frac{(1+e^{-y})}{ x}$ .I have the graph for the problem but need to know how to obtain it manually.

Question

The problem here is not to show the graph but if or how (i.e if it can be plotted) can I obtain it roughly without the use of a graph generator . Any help will be appreciated .

Answer 1

a good start is to graph the function:

$x = \frac{(1+e^{-y})}{y^2}$. Note that the $y^2$ in the denominator creates an asymptote at $y=0$ and more importantly, since it's an even power, the function will wrap towards the asymptote on both sides (odd powers have opposing wraps).

As for the other asymptote, you find it by taking the limit going to positive and negative infinity. Recall that $$\lim_{y \to \infty} e^{-y} = 0$$ and $$\lim_{y \to -\infty} e^{-y} = \infty $$ Therefore, taking the positive and negative limits of the function will allow you to see the asymptotic behavior of the function analytically:

$$\lim_{y \to \infty} \frac{1+e^{-y}}{y^2}=0$$ $$\lim_{y \to -\infty} \frac{1+e^{-y}}{y^2} = \infty$$ You can check both of those via calculus. This implies we have a vertical asymptote at $x=0$ since that's the value y approaches. To summarize, forcing your functions to be represented as either $f(x)$ or $f(y)$ can be very useful and from there checking various properties from rational functions or whatever parent functions are present is a good start. Normally, we would also look for holes, x/y intercepts as well (or slant asymptotes in some cases) but I have omitted those since this function doesn't have them.

Good Luck!