Question

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

Answer 1

Graph of `y=(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.
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Make a table of value of `f(x)` 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)

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Before to start to draw, we can observe some things on `f(x)`
Key point of `f(x)`:
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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!=0 <=> (x-2)*(x+2)!=0 <=> x!=2` & `x!=-2`

We name the two straight lines `x=2` and `x=-2`, vertical asymptotes of `f(x)`, ie, that the curve of `f(x)` never crosses this lines.
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Root of `f(x)` :

`f(x)=0 <=> x^3+1=0<=>x=-1`

Then :`(-1,0) in C_f`

Note : `C_f` is the representative curve of `f(x)` on the graph
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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!