Sketching graphs : Most importaint points

Some key points:

• Zeroes

• Local maxima/minima

• Boundary/ies of the domain, if the domain is bounded from at least one direction

• Boundary/ies of the support (where the function is non-zero), if the support is bounded in at least one direction

• Asymptotes, if such exist

• Points of inflection

Application:

$$y=\frac{1-x-x^2}{x^2}.$$

First, let's look for zeroes: $1-x-x^2=0$ when $$x=\frac{-1\pm \sqrt 5}2.$$

So estimate $\sqrt 5$ and use that, perhaps noting the exact values.

Next, we can rewrite this as $$y=\frac{1-x}{x^2}-1.$$ As $x$ increases or decreases without bound, this whole thing approaches $y=-1$, so draw that line.

$$y'=\frac{-x^2+2x(x-1)}{x^4}=\frac{x-2}{x^3},$$ which is $0$ exactly at $2$.

$$y''=\frac{x^3-3x^2(x-2)}{x^6}=\frac{-2x+6}{x^4},$$ which is positive at $2$, so $x=2$ is a local minimum. Draw that.

The function is undefined at $x=0$ and in fact has no limit. Figure out which ways it goes.