Problem 21 - College Algebra

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=-2 x^{4}-3 x^{2}+x-1 $$

Step-by-Step Solution

Answer

The polynomial \(-2x^4 - 3x^2 + x - 1\) has both ends pointing downward; \(f(x) \to -\infty\) as \(x \to +\infty\) and \(x \to -\infty\).

Step 1: Identify the Leading Term

The leading term of the polynomial is the term with the highest power of \(x\). In \(f(x) = -2x^4 - 3x^2 + x - 1\), the leading term is \(-2x^4\).

Step 2: Determine the Leading Coefficient

The leading coefficient is the constant multiplying the leading term. For the term \(-2x^4\), the leading coefficient is \(-2\).

Step 3: Analyze the Degree of the Polynomial

The degree of a polynomial is the highest power of \(x\). Here, the degree is 4, meaning it is an even degree polynomial.

Step 4: Determine End Behavior Based on Degree and Leading Coefficient

For an even-degree polynomial, the end behavior is determined by the sign of the leading coefficient. If the leading coefficient is negative, as in \(-2x^4\), both ends of the graph will point downward as \(x\) approaches positive or negative infinity.

Step 5: Write the Formal End Behavior

Using the above information, write the end behavior as: 1. As \(x \to +\infty\), \(f(x) \to -\infty\).2. As \(x \to -\infty\), \(f(x) \to -\infty\).

Key Concepts

Leading Term of Polynomial

In any polynomial, the leading term is the part with the highest power of the variable "x". This term tells you a lot about the overall behavior of the polynomial, especially as the values of "x" become very large or very small. For the polynomial given by the expression \(f(x) = -2x^4 - 3x^2 + x - 1\), the leading term is \(-2x^4\). This term is significant because its degree will help us determine the polynomial's degree and its leading coefficient will help ascertain the end behavior.

Leading Coefficient

The leading coefficient is simply the coefficient or number in front of the leading term in a polynomial, which is also known as the term with the highest power. In the polynomial function \(f(x) = -2x^4 - 3x^2 + x - 1\), the leading term is \(-2x^4\), and thus the leading coefficient is \(-2\). The sign and magnitude of this coefficient help us understand the end behavior of the polynomial, particularly whether the graph points upwards or downwards as "x" moves towards positive or negative infinity.

Degree of Polynomial

The degree of a polynomial is determined by the highest power of "x" present in the expression. In the context of the polynomial \(f(x) = -2x^4 - 3x^2 + x - 1\), the highest power of "x" is \(4\), which classifies it as a 4th-degree polynomial. Knowing the degree is crucial because it influences the shape and end behavior of the graph. The degree indicates the maximum number of roots and the concavity or general shape of the graph over various ranges of "x" values.

Even-Degree Polynomial

An even-degree polynomial, such as the one in our example with a degree of 4, has certain predictable end behavior characteristics. Polynomials with an even degree tend to have their graph behaviors mirror each other at both ends. This means if one end of the graph is pointing down, the other end will also be pointing down, and vice versa for upward direction. With our example function \(f(x) = -2x^4 - 3x^2 + x - 1\), because the leading coefficient is negative, both ends of the graph head downward as \'x\' approaches both positive and negative infinity.