Chapter 5.4, Problem 10MCQ

To determine

Explain the end behavior, odd-degree or an even-degree polynomial function and state the number of real zeros using graph.

Answer to Problem 10MCQ

The graph represents the odd-degree polynomial function and it has three real zeros.

Explanation of Solution

Given:

The given graph is

  

Calculation:

The end behavior is the behavior of the graph of $f\left(x\right)$ as $x$ approaches positive infinity $\left(x\to +\infty \right)$ or negative infinity $\left(x\to -\infty \right)$ . The degree and leading coefficient of a polynomial function determine the end behavior of the graph.

The shape of the graph of the polynomial function shows the maximum number of times the graph of the function intersect the x-axis. This is the same number as the degree of the polynomial. The number of times a graph crosses the x-axis equals the number of real zeros.

Odd-degree function will always have an odd number of real zeros and even-degree function will always have an even number of real zeros or no real zeros at all.

In the given graph

  

  $\begin{array}{l}f\left(x\right)\to +\infty \text{as}x\to -\infty \\ f\left(x\right)\to -\infty \text{as}x\to +\infty \end{array}$

Since the end behavior is in opposite directions, it is an odd-degree polynomial function.

The graph intersects the x-axis at three points, so there are three real zeros.