Chapter 3, Problem 2RCC

To determine

To sketch: The graph of all the possible end behavior of polynomials of odd degree and of even degree.

Explanation of Solution

The end behavior of the polynomial is determined by the degree n and the sign of the leading coefficient.

The end behavior of a polynomial describes the behavior of the graph of function at the end of the x-axis.

For positive leading coefficient and odd degree the end behavior is,

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

The graph is below

Figure (1)

Figure (1) shows the end behavior of graph of polynomial with positive leading coefficient and odd degree.

For negative leading coefficient and odd degree the end behavior is,

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

The graph is below,

Figure (2)

Figure (2) shows the end behavior of graph of polynomial with negative leading coefficient and odd degree.

For positive leading coefficient and even degree the end behavior is,

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

The graph is below

Figure (3)

Figure (3) shows the end behavior of graph of polynomial with positive leading coefficient and odd degree.

For negative leading coefficient and even degree the end behavior is,

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

The graph is below,

Figure (4)

Figure (4) shows the end behavior of graph of polynomial with negative leading coefficient and even degree.