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To me it looks like a reciprocal function mixed with a cubic function. I tried out x/((x+1)(x-4)) and it looks very close, but I can't figure out the correct transformations to apply. What is the best way to go about trying to find an equation for this?

Timmy Abina
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  • It's hard to tell without more exact points than $(2,0)$ and $(0,2)$ and the asymptotes, but at least you have to change the numerator to $4(x-2)$ to match these. – zwim Mar 22 '23 at 14:28
  • Consider the partial fraction form. It's like $ A / (x+1) + B /(x-4) + C(x)$. What is the polynomial $C(x)$? How can we determine $A$ and $B$? – Calvin Lin Mar 22 '23 at 14:29

1 Answers1

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Based on your answer, our function should take the following form:

$$f(x)=\frac{a(x-b)}{(x+1)(x-4)}$$

Since $f(2)=0$, then $b=2$.

$$\implies f(x)=\frac{a(x-2)}{(x+1)(x-4)}$$

Since $f(0)=2$, then $a=4$. Therefore,

$$f(x)=\frac{4(x-2)}{(x+1)(x-4)}$$

Leonidas Lanier
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    Also worth mentioning that the horizontal asymptote (looks like) is $ y = 0$, so there are no quadratic or higher terms in the numerator. – Calvin Lin Mar 22 '23 at 14:32
  • That's correct--even quadratic terms can be outruled. – Leonidas Lanier Mar 22 '23 at 14:33
  • And as an aside, the denominator could even be of the form $(x+1)(x-4)^3$, which we can't necessarily rule out unless we have much more detail about the graph. – Calvin Lin Mar 22 '23 at 14:34