Suppose we want to find the end behavior of
\(f(x)=3(4x+7)(x-2)^3(x+1)^4.\text{.}\) In this case, we have all the parts multiplied together, rather than added. We need it to be added in order to quickly find the leading term. Theoretically, we could multiply it all out, but my goodness, that would be way too much work. Donβt do that. If we needed to know the entire formula, then that would be our only choice. But we donβt need to know the entire formula. We only need to know the part with the highest power on
\(x\text{.}\)
In order to find the leading term, we just need to multiply the leading term of each piece:
\begin{equation*}
3(4x)(x)^3(x)^4=12x^8
\end{equation*}
Therefore, our leading term is \(12x^8\text{.}\) Therefore, because our leading coefficient is positive and our degree is even, our end behavior is
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As \(x\rightarrow \infty\text{,}\) \(y \rightarrow \infty\)
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As \(x\rightarrow -\infty\text{,}\) \(y \rightarrow \infty\)