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First, apologies about the basic question, I'm a first time Calc I student with no prior calc experience. I'm looking for the general steps of how to sketch a graph given a basic function like $(1/16)x^4$ - $12x^2$ + 4.

I understand that you need to find the first derivative and set it equal to 0 to find the critical points, and then find the second derivative and do likewise to it to find the inflection points. But how do I find the general shape of the graph, like if the line is going up toward infinity or negative infinity between points?

Thanks for your help

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  • $\begingroup$ For large positive or negative $x$, a polynomial behaves approximately like its leading term ($\frac1{16}x^4$ in this case). Since it is of even degree and the leading coefficient is positive, that means it approaches $+\infty$ as $x\to \infty$ and as $x\to-\infty$. $\endgroup$ – MPW Dec 7 '15 at 17:12
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But how do I find the general shape of the graph, like if the line is going up toward infinity or negative infinity between points?

  • Look for the variable which has the highest power. That will most probably be the one which will determine where the value of the function tends to as $x\rightarrow\infty$ and $x \rightarrow -\infty$.

  • If you could factorize the function, consider the points for which the function is not defined (denominator), and use limits from approaching from $+$ and $-$ of it. This will give the direction whether it is $+\infty$ or $-\infty$.

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