Problem 22 - College Algebra

Question

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{3}-6 x^{2}+x+3$$

Step-by-Step Solution

Answer

The end behavior of the graph of the given polynomial function is: it falls to the left and rises to the right.

Step 1: Identify the leading coefficient and the degree

The leading coefficient is the coefficient of the term with the highest power. So here in the polynomial \(f(x)=11x^3-6x^2+x+3\), the leading coefficient \(a\) is 11. The highest power in the polynomial is 3, which is our degree \(n\).

Step 2: Apply the Leading Coefficient Test

According to the Leading Coefficient Test, for a polynomial of odd degree like 3, if the leading coefficient is positive which it is in our case (11), the graph will fall to the left and rise to the right.

Key Concepts

Polynomial Functions

A polynomial function is an expression constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions are represented in the form of eq{f(x)=a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0}where the coefficients (eq{a_n, a_{n-1}, ..., a_0} )are real numbers, and the variable x is raised to whole number powers. The term with the highest power of x determines the polynomial's degree and the leading coefficient.
Understanding polynomial functions is crucial, as they are versatile for modeling different physical phenomena and solving real-life problems. Characteristics such as the number of roots, the possible number of turning points, and the end behavior of the graph are all determined by the polynomial's degree and leading coefficient.

End Behavior of Graphs

The end behavior of a graph describes the direction in which the graph tends as it moves towards infinity or negative infinity along the x-axis. This characteristic is particularly important for understanding the long-term trends of polynomial functions.
By applying the Leading Coefficient Test, which analyzes the sign and degree of the leading coefficient, we can predict end behavior. For instance, a polynomial function with a positive leading coefficient and an odd degree will have a graph that falls to the left and rises to the right. Conversely, with a negative leading coefficient and an odd degree, the graph will rise to the left and fall to the right.

Visualization

Imagine drawing a line that extends from the graph towards both ends of the x-axis—it's the direction of these lines that gives us the end behavior. End behavior provides valuable insight into the function's global behavior, beyond just the visible part of its plot.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable x that appears in the function. It is an essential quality because it offers information about many aspects of the polynomial, including the end behavior of its graph and the maximum number of solutions or roots the polynomial can have.
For example, a polynomial of degree 3, called a cubic polynomial, can have up to 3 real roots and up to two turning points. The degree of the polynomial is directly related to these characteristics because it shapes the polynomial's curve. A higher-degree polynomial will generally have more complexity in its shape. The Leading Coefficient Test uses the degree, along with the leading coefficient, to predict how the graph will behave at the extremities of the x-axis.

• Even-degree polynomials have the same end behavior on both sides of the graph.
• Odd-degree polynomials have opposite end behaviors on each side of the graph.

This information is crucial for sketching the graph of a polynomial function and for understanding its general structure.