Chapter 3.2, Problem 42E

To determine

The end behavior of $P$ and compare the graphs of $P$ and $Q$ in large and small viewing rectangles.

Answer to Problem 42E

The end behavior of P is $y\to -\infty$ as $x\to \infty$ and $y\to \infty$ as $x\to -\infty$ , both the graphs look alike in large viewing rectangle, and both the graphs quite separated in small viewing rectangle.

Explanation of Solution

Given information:

The given functions are $P\left(x\right)=-\frac{1}{8}{x}^3+\frac{1}{4}{x}^2+12x$ and $Q\left(x\right)=-\frac{1}{8}{x}^3$ .

Calculation:

Since $P$ has an odd degree and a negative coefficients. So, it has the following end behavior.

  $y\to -\infty$ as $x\to \infty$ and $y\to \infty$ as $x\to -\infty$

The graph of the functions $P\left(x\right)=-\frac{1}{8}{x}^3+\frac{1}{4}{x}^2+12x$ and $Q\left(x\right)=-\frac{1}{8}{x}^3$ in large viewing rectangles is as follows.

  

As shown in the graph, both the graphs look alike in large viewing rectangle which means the behavior is same for both polynomials.

The graph of the functions $P\left(x\right)=-\frac{1}{8}{x}^3+\frac{1}{4}{x}^2+12x$ and $Q\left(x\right)=-\frac{1}{8}{x}^3$ in small viewing rectangles is as follows.

  

As shown in the graph, both the graphs quite separated in small viewing rectangle.

Therefore, the end behavior of P is $y\to -\infty$ as $x\to \infty$ and $y\to \infty$ as $x\to -\infty$ , both the graphs look alike in large viewing rectangle, and both the graphs quite separated in small viewing rectangle.