# How do you graph f(x)= (x^3+1)/(x^2-4)?

Jul 12, 2015

Graph of $y = \frac{{x}^{3} + 1}{{x}^{2} - 4}$
graph{(x^3+1)/(x^2-4) [-40, 40, -20,20]}

#### Explanation:

There is no secret to graph a function.

Make a table of value of $f \left(x\right)$ and place points.
To be more accurate, take a smaller gap between two values of $x$

Better, combine with a sign table, and/or make a variation table of f(x). (depending on your level)



Before to start to draw, we can observe some things on $f \left(x\right)$
Key point of $f \left(x\right)$:



Take a look to the denominator of the rational function : ${x}^{2} - 4$

Remember, the denominator can't be equal to $0$

Then we will be able to draw the graph, when :

${x}^{2} - 4 \ne 0 \iff \left(x - 2\right) \cdot \left(x + 2\right) \ne 0 \iff x \ne 2$ & $x \ne - 2$

We name the two straight lines $x = 2$ and $x = - 2$, vertical asymptotes of $f \left(x\right)$, ie, that the curve of $f \left(x\right)$ never crosses this lines.


Root of $f \left(x\right)$ :

$f \left(x\right) = 0 \iff {x}^{3} + 1 = 0 \iff x = - 1$

Then :$\left(- 1 , 0\right) \in {C}_{f}$

Note : ${C}_{f}$ is the representative curve of $f \left(x\right)$ on the graph




N.B : J'ai hésité à te répondre en français, mais comme nous sommes sur un site anglophone, je prefère rester dans la langue de Shakespeare ;) Si tu as une question n'hésite pas!