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Sketch the graph of the function $y=x(4-x)-83\ln(x)$. Indicate the transition points (local extrema and points of inflection).

I understand how to solve this problem mathematically, however, is there any way of analyzing the question to get a better understanding of what the graph would look like? I am not the best when it comes to understanding the appearance of graphs via their function.

CaptainObvious
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Mone Skratt Henry
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1 Answers1

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Some general strategies:

  • If not explicitly stated, figure out the domain. For our example, since the logarithm is only defined for positive $x$, our domain is $(0, \infty)$.
  • Get the asymptotes of the graph. For our example, as $x \to 0^+, x(4-x) \to 0$ and $\ln x \to -\infty$. Hence $y \to +\infty$ and $x=0$ is an asymptote.
  • Locate stationary points (local extrema and stationary points of inflection) by differentiation. For our example it can be proven that there are no stationary points by using the quadratic discriminant.
  • Analyze the behavior of the graph at $\pm \infty$. For our example, only $+\infty$ is applicable because of the domain. As $x \to + \infty, x(4-x) \to -\infty$ and $\ln x \to + \infty$ so $y \to -\infty$.
  • Use continuity property if applicable. Our graph is made up of a polynomial ($x(4-x)$) and a logarithm function, both of which are continuous. Hence our graph will be continuous (and in fact, "smooth"). Using this property in addition to the information above is usually sufficient to determine the general shape of the graph.

And finally, when in doubt, it never hurts to locate a few points on the graph (by substituting a few numbers into the equation).

Kelvin Soh
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