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Curve Sketching

Here we continue the work that we’ve done in Increasing or Decreasing?, and Intervals of Concavity. If you haven’t read through those yet, I would suggest going back and reading them since we will be using the information that we found there. The next step will now be to put the information together so that we can sketch the graph of the function.

Graph the function

Graph the function \(f(x)=x(x-2)^{3}\) labeling all pertinent information.

Before we start graphing, we want to understand what we should be highlighting. In particular, the problem asks us to label all pertinent information, but what does that mean? The idea is that if you were to look at a graph, what would stick out to you? I would suggest you take a moment to think about this, but the things that will catch my eye are the following.

  • From the function
    • Points of discontinuity, including vertical asymptotes.
    • Horizontal asymptotes.
    • Zeroes of the function (that is \(x\)-intercepts).
    • \(y\)-intercepts.
  • From the derivative.
    • Local extrema (critical points).
    • Intervals of increasing and decreasing.
  • From the second derivative.
    • Inflection points.
    • Intervals of concavity.

Function information

As we look at the function, our goal is to find things that will stick out about heights. The first thing that would stick out, is if there were no function value. In this case, however, we know that \(f(x)=x(x-2)^{3}\) is a polynomial, therefore, the function is continuous over all real numbers. Hence, we will have no points of discontinuity.

Next, we would want to determine if the function values tended toward a specific value as \(x\) got arbitrarily large in the positive or negative direction. However, if we let \(x \to \infty\) we note that \(f(x) \to \infty\) and if \(x \to -\infty\) we find \(f(x) \to \infty\). Therefore, in both directions, the function will grow without bound giving us no horizontal asymptotes.

Now we need to find zeroes of the function. In this case we set
\begin{align*}
f(x)&=0 \\
x(x-2)^{3}&=0 \\
x&=0 \text{ or } (x-2)^{3}=0 \\
x&=0 \text{ or } x=2.
\end{align*}
We, therefore, find that the zeroes of the function are at \(x=0\) and \(x=2\).

Lastly, we may want to know the \(y\)-intercept. That is, the \(y\)-value when \(x=0\). In this case, we find that \(f(0)=0\).

Derivatives

In Increasing or Decreasing? we found the information regarding critical points and intervals of increasing and decreasing. We summarize this using the number line below.

Furthermore, for the critical points we had a local minimum at \((\frac{1}{2}, -\frac{27}{16})\) and local maximum at \((2,0)\).

In Intervals of Concavity we found the information regarding inflection points and concavity. We summarize this using the number line below.

Also, the inflection points were \((1,-1)\) and \((2,0)\).

Combining derivatives

As we continue to work toward our graph, we want to combine the information regarding the shape of the graph given by both the first and second derivatives. Since we want to graph, we will begin by looking at what a graph of a function with each of the properties would be.

  • Increasing.
  • For an increasing function, the \(y\)-values gets larger as the \(x\)-value gets larger. That is, the \(y\) goes up as the \(x\) goes right. In this case, I imagine my son, Arthur trying to walk up a slide.
  • On the other hand, a decreasing function will have the \(y\)-values going down as the \(x\)-values go right. Here, I imagine Arthur going down the slide.
  • Then, if we have that if the function is concave up, it will have the shape of bowl pointed up. As a personal note, I imagine a bowl with cereal in it. I happen to really like Cinnamon Toast Crunch, and in this case, I get to eat the cereal out of my bowl. This picture would look like,
  • In the case that we have a concave down function we get that the function will look bowl shape down. In this case, I image my son Arthur took the cereal and dumped the bowl upside down. Then I don’t get to eat my cereal, which is sad. 🙁
  • If we combine these together, we get four combinations of increasing or decreasing and concave up and down. Doing so, we arrive at the following shapes.
  • Now that we know what the derivatives tell us about the shape, we can create a shape number line similar to what we did for each derivative. In this case, we make a partition number for each partition number of the first and second derivative. Therefore, we break the number line at the points \(x=\frac{1}{2},1\) and \(2\). We then find that
    • On \((-\infty,\frac{1}{2})\) \(f(x)\) is decreasing and concave up.
    • On \((\frac{1}{2},1)\) \(f(x)\) is increasing concave up.
    • On \((1,2)\) the function is increasing concave down.
    • On \((2, \infty)\) the function is increasing concave up.
  • We can summarize this with the number line below.
  • Sketching

  • The last thing we need to do is to actually graph the function. We begin by plotting any points of interest. This includes the intercepts, critical points and inflection points. Here we will have,
  • Now that we have plotted the points of interest, we can now play connect the dots. However, as we connect the dots, we want to make sure that we use the appropriate shapes for each of the intervals. Since we found the shapes above, we can graph this as
  • Conclusion

  • I hope this helped provide you with an outlining of how to graph functions by hand using curve sketching techniques. If it did, make sure to like the post and share it on Social Media.

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