Chapter 3: Polynomial Functions and Their Graphs – College Algebra Study Notes

Polynomial and Rational Functions

Objectives

This section introduces the fundamental concepts related to polynomial functions and their graphs. Students will learn to identify polynomial functions, analyze their graphical characteristics, determine end behavior, find zeros and their multiplicities, apply the Intermediate Value Theorem, and understand the relationship between degree and turning points.

• Identify polynomial functions

• Recognize characteristics of graphs of polynomial functions

• Determine end behavior

• Use factoring to find zeros of polynomial functions

• Identify zeros and their multiplicities

• Use the Intermediate Value Theorem

• Understand the relationship between degree and turning points

• Graph polynomial functions

Definition of a Polynomial Function

Key Concepts

A polynomial function is defined as follows:

• Let n be a nonnegative integer and let the coefficients be real numbers.

• The general form of a polynomial function is: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ where $a_n \neq 0$.

• The degree of the polynomial is n.

• The leading coefficient is $a_n$, the coefficient of the highest power of $x$.

Example: $f(x) = 2x^3 - 5x^2 + x - 7$ is a polynomial of degree 3 with leading coefficient 2.

Graphs of Polynomial Functions – Smooth and Continuous

Characteristics of Polynomial Graphs

Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.

• Smooth: The graph contains only rounded turns, with no sharp corners.

• Continuous: The graph has no breaks and can be drawn without lifting your pencil from the paper.

Example: The graph of $f(x) = x^3 - 2x^2 + x$ is smooth and continuous.

End Behavior of Polynomial Functions

Understanding End Behavior

The end behavior of a function describes how its graph behaves as $x$ approaches positive or negative infinity.

• Although the graph may have intervals of increase or decrease, it will eventually rise or fall without bound as $x$ moves far to the left or right.

• The sign of the leading coefficient $a_n$ and the degree $n$ determine the end behavior.

Example: For $f(x) = -x^4 + 3x^2$, as $x \to \pm\infty$, $f(x) \to -\infty$.

Leading Coefficient Test

Determining End Behavior

The Leading Coefficient Test helps predict the end behavior of a polynomial function based on its degree and leading coefficient.

• If the degree $n$ is odd:

  • If $a_n > 0$, the graph falls to the left and rises to the right.

  • If $a_n < 0$, the graph rises to the left and falls to the right.

• If the degree $n$ is even:

  • If $a_n > 0$, the graph rises to both the left and right.

  • If $a_n < 0$, the graph falls to both the left and right.

Degree ($n$)	Leading Coefficient ($a_n$)	End Behavior
Odd	Positive	Falls left, rises right
Odd	Negative	Rises left, falls right
Even	Positive	Rises left and right
Even	Negative	Falls left and right

Example: For $f(x) = 2x^3$, degree is odd and leading coefficient is positive, so the graph falls left and rises right.

Example: Using the Leading Coefficient Test

Application

Apply the Leading Coefficient Test to determine the end behavior of a polynomial function.

• Given: $f(x)$ has degree 4 (even) and leading coefficient 1 (positive).

• Conclusion: The graph rises to the left and to the right.

Example: The graph of $f(x) = x^4 - 2x^2 + 1$ rises on both ends.