Problem 28 - College Algebra

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=x^{3}+27 $$

Step-by-Step Solution

Answer

As \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 1: Understand the Polynomial Degree

The function given is a polynomial function, specifically a cubic polynomial of the form \( f(x) = x^3 + 27 \). The highest degree of \( x \) is 3, which is the leading term.

Step 2: Determine Leading Coefficient and Degree

The leading coefficient of the polynomial is the coefficient of \( x^3 \), which is 1, and the degree of the polynomial is 3, which is odd. This influences the end behavior of the function.

Step 3: Identify End Behavior Based on Leading Term

For a polynomial \( ax^n \) where \( a > 0 \) and \( n \) is an odd degree, as \( x \to \pm\infty \), the end behavior of the function \( f(x) \) will show that \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that combine numbers and variables using algebraic operations. They are one of the most familiar types of functions in algebra. Any polynomial can be written in the form:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer.

• The highest power of \( x \) in the polynomial is called the degree of the polynomial.
• The coefficients, like \( a_n \), influence the width and direction of the graph.

Each polynomial function has a unique graph, and the degree of the polynomial helps in determining its shape and end behavior. The variety in their structure makes polynomial functions valuable tools for modeling real-world situations.

Cubic Polynomial

A cubic polynomial is a specific type of polynomial that has a degree of three. It is called 'cubic' because its highest term is raised to the power of three. The general form of a cubic polynomial is:\[ f(x) = ax^3 + bx^2 + cx + d \]where \( a, b, c, \) and \( d \) are constants, and importantly, \( a eq 0 \).Characteristics of cubic polynomials:

• The graph typically has one inflection point, where it changes curvature.
• Its end behavior is influenced by the leading term \( ax^3 \).
• If the leading term is positive \( (a > 0) \), the ends of the graph rise to the right and fall to the left.
• If the leading term is negative \( (a < 0) \), the ends of the graph fall to the right and rise to the left.

Cubic polynomials often model systems with a single significant curvature change, such as some economic or physics problems.

Leading Coefficient and Degree

The leading coefficient and degree of a polynomial are crucial in determining its overall shape, especially for the end behavior. The leading coefficient is the coefficient of the term with the highest degree, while the degree is the highest power of \( x \) in the polynomial.

• For example, in \( f(x) = x^3 + 27 \), the leading coefficient is 1, and the degree is 3.
• The degree tells us how many roots the polynomial might have and provides insight into the end behavior.
• The leading coefficient affects the stretching or compressing of the graph as well as which direction it opens.

Understanding these two components allows us to predict the behavior of polynomials without graphing them.

Odd Degree Polynomials

Odd degree polynomials, such as a cubic polynomial, have unique characteristics on their graphs, particularly regarding end behavior. An odd degree means the highest exponent of the polynomial is an odd number (1, 3, 5, etc.).Properties of odd degree polynomials:

• End behavior is linked to the sign of the leading coefficient.
• If the leading coefficient is positive, the graph rises on the right and falls on the left.
• If the leading coefficient is negative, the graph falls on the right and rises on the left.
• Odd degree polynomials cross the x-axis at least once, meaning they have at least one real root.

For the function \( f(x) = x^3 + 27 \), the odd degree of 3 and a positive leading coefficient suggest that as \( x \to \pm \infty \), the graph will go from negative to positive, creating an S-like shape.