Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=-x^{4} $$

Step-by-Step Solution

Answer

The function \( f(x) = -x^4 \) decreases to \(-\infty\) as \( x \to \infty \) and \( x \to -\infty \).

Step 1: Analyze the Function's Leading Term

The function given is \( f(x) = -x^4 \). The leading term is \( -x^4 \). This term is the highest degree term of the polynomial and determines the end behavior of the function as \( x \to \infty \) or \( x \to -\infty \).

Step 2: Determine the Degree and Leading Coefficient

The degree of the function is 4, which is even. The leading coefficient is -1, which is negative. These characteristics will dictate the end behavior of the polynomial function.

Step 3: Determine the End Behavior based on Polynomial Characteristics

For any polynomial function \( ax^n \) where \( n \) is even and \( a \) is negative, as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \). Therefore, both ends of the graph point downwards.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables, coefficients, and the operation of addition, subtraction, and multiplication. These functions are expressed in the form of a sum of terms, such as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where:

• \( n \) is a non-negative integer, known as the degree of the polynomial.
• \( a_n, a_{n-1}, ..., a_0 \) are the coefficients, and \( a_n eq 0 \).

This structure allows polynomial functions to model a wide range of scenarios and exhibit varied behaviors depending on their degree and coefficients. They are continuous and smooth, meaning they don't have any breaks, holes, or sharp turns. Understanding the form of a polynomial function helps predict the end behavior, which is how the function behaves as \( x \to \infty \) or \( x \to -\infty \). In this exercise, the function \( f(x) = -x^4 \) is a polynomial function of degree 4.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial function. It significantly influences the polynomial's behavior, particularly its end behavior.Consider the polynomial function \( f(x) = -x^4 \). Here, the leading coefficient is -1, since the highest degree term is \( -x^4 \).

• Positive leading coefficient: If the leading coefficient is positive and the degree is even, the ends of the graph will both point upwards.
• Negative leading coefficient: If the leading coefficient is negative and the degree is even, both ends of the graph will point downwards, as seen in our example.

For odd-degree polynomials, the leading coefficient affects whether the graph rises or falls as \( x \to \infty \) or \( x \to -\infty \). Understanding the leading coefficient helps determine the general shape of the graph, providing insight into the overall behavior of the function.

Degree of Polynomial

The degree of a polynomial is a crucial factor in understanding its graph's end behavior. It is determined by the highest power of the variable in the polynomial expression.For the polynomial \( f(x) = -x^4 \), the degree is 4.

• Even-degree polynomials: These have the same end behavior at both extremes of the graph. For instance, if the leading coefficient is negative, like in our exercise, as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).
• Odd-degree polynomials: These have opposite end behaviors at each end of the graph. For example, a positive leading coefficient suggests that \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \), and vice versa for a negative leading coefficient.

Understanding the degree of a polynomial is vital for predicting how the graph will behave at its extremes, assisting in sketching or analyzing the function's general shape.