Chapter 1.4, Problem 28E

To determine

Observe the graphs.

Answer to Problem 28E

Misleading impression of the function.

Explanation of Solution

Given information:

Graph the polynomials $P(x)=3x^5-5x^3+2x$ and $Q(x)=3x^5$ on the same screen, first using the viewing rectangle $[-2,2]$ by $[-2,2]$ and then changing to $[-10,10]$ by $[-10,000,10,000]$ .What do you observe from these graphs?

Calculation:

Consider the polynomials $P(x)=3x^5-5x^3+2x$ and $Q(x)=3x^5$ .

Figure(a) shows the graph of $P(x)=3x^5-5x^3+2x$ and $Q(x)=3x^5$ produced by a graphing calculator using the viewing rectangle $[-10,10]$ by $[-10,000,10,000]$ . At first glance, the graph appears to be reasonable, and it looks like same as shown in the following figure:

  

But if we change the viewing rectangle to be ones shown in the following figure(b), the graph looks very different.

  

Now we see that what went wrong in figure(a), the oscillations of $P(x)=3x^5-5x^3+2x$ are so rapid that when the calculator plots points and joins them, it misses most of the maximum minimum points and gives a very misleading impression of the graph.

Hence, we observe from a figure(a) and figure (b), if we use an inappropriate viewing rectangle can give a misleading impression of the graph of a function.