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I need to be able to find an equation from this graph

graph_goal

So far I have this graph with the equation $-1/((x-3)^3)$

graph_so_far

I can see from the desired graph that there is no horizontal asymptote, compared to the graph I currently have. I'm lost on what to do next.

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1 Answer

Examining the graph, you can see that it crosses the $x$ axis at $-2$ and $8$. From that, It must be something like $(x-8)(x+2)\cdot a$.

Further inspection shows the $x=3$ asymptote. So your "$a$" must be something like $1/(x-3)$.

So, your graph should come from the equation $(x^2 -6x -16)/(x-3)$.

EDIT: Forgot about something. Your graph can be changed by a multiplier $n$, being $n\gt0$, in this way: $$n\bigg(\frac{x^2-6x-16}{x-3}\bigg)$$ So, your graph can be changed by any multiplier, being it $\frac 1 2$ , $2$ or even $1$ (in this last case keeping the original answer). The roots will be kept ($-2$ and $8$, in this case) and the asymptote will be kept as well. So, a first look may not determine exactly which equation gives that graph, but it might get a clue to what the $n$ can be if further inspected.

Edit (2): As seen on the graph, the point at $x=4$ gives out $y=-24$. So, if the grid on the graph is 1 unit of $x$ by 1 unit of $y$, your "$n$" stated above must be $1/12$, giving the point ($4$,$-2$). So, your final answer should be $$y=\frac{1}{12}\bigg(\frac{x^2 -6x-16}{x-3}\bigg)$$

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