Problem 23 - College Algebra

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$ f(x)=5 x^{4}+7 x^{2}-x+9 $$

Step-by-Step Solution

Answer

The graph of the polynomial function \(f(x)=5 x^{4}+7 x^{2}-x+9\) opens upwards on both ends. As x approaches positive or negative infinity, f(x) approaches positive infinity.

Step 1: Identify the leading term and its coefficient

Here, the leading term of the polynomial function \(f(x)=5 x^{4}+7 x^{2}-x+9\) is \(5x^4\) and the leading coefficient is 5.

Step 2: Identify the degree of the polynomial

The degree of a polynomial is determined by the highest power of x in its terms. In the polynomial function \(f(x)=5 x^{4}+7 x^{2}-x+9\), the highest power of x is 4. So, the degree of the polynomial is 4 which is an even-numbered degree.

Step 3: Apply the Leading Coefficient Test

The Leading Coefficient Test states that if the leading coefficient of a polynomial is positive and the degree of the polynomial is even, the graph opens upwards on both ends. Here, the leading coefficient of the polynomial function is 5 (a positive number), and the degree is 4, an even number. Hence, according to the Leading Coefficient Test, the graph of the given polynomial opens upwards on both ends. The end behavior of the polynomial is, as x goes to infinity, f(x) goes to infinity and as x goes to negative infinity, f(x) also goes to infinity.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of one or more variables, each multiplied by coefficients. It takes the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where the \( a_i \)s are constants. In simpler terms, a polynomial consists of terms where each term includes a coefficient, a variable base (like \( x \)), and an exponent which is a non-negative integer.
If we consider the polynomial \( f(x) = 5x^4 + 7x^2 - x + 9 \), it is composed of several terms: \( 5x^4, 7x^2, -x, \) and \( 9 \).

• The coefficients here are 5, 7, -1, and 9.
• The powers of \( x \) are 4, 2, 1, and 0 respectively.

This structure allows us to classify it as a polynomial.

End Behavior

The end behavior of a polynomial function describes what happens to the values of \( f(x) \) as \( x \) approaches positive or negative infinity. Understanding the end behavior helps us determine the direction in which the arms of the graph extend.
In the Leading Coefficient Test, the end behavior is examined by looking at:

• The sign of the leading coefficient
• Whether the degree is even or odd

For the polynomial \( f(x) = 5x^4 + 7x^2 - x + 9 \), we determine that:

• If the leading coefficient is positive and the degree is even, the polynomial graph rises on both ends.
• If the leading coefficient was negative and the degree even, the graph would fall on both ends.
• For degrees that are odd and positive leading coefficients, the graph falls left and rises right, and vice versa for negative coefficients.

In our example, because the leading coefficient is positive and the degree is even, both ends of the graph rise.

Degree of the Polynomial

The degree of a polynomial is defined by the highest exponent of the variable \( x \) in the polynomial expression. It determines many properties of the polynomial, including the maximum number of roots and turns the graph may have.
For example, in \( f(x) = 5x^4 + 7x^2 - x + 9 \), the highest exponent appears in the term \( 5x^4 \). Therefore, the degree of this polynomial is 4, which is an even number.
Knowing the degree is crucial as it dictates:

• The possible number of zeroes the polynomial can have.
• The maximum number of turning points is \( n-1 \), where \( n \) is the degree.

Evenness or oddness of the degree significantly affects the graph's end behavior.

Leading Term

The leading term of a polynomial is the term that contains the highest power of the variable \( x \). It is very significant because it largely influences the behavior of the polynomial, especially as \( x \) increases or decreases without bound.
For the polynomial \( f(x) = 5x^4 + 7x^2 - x + 9 \), the leading term is \( 5x^4 \). This term informs us that:

• The degree of the polynomial is 4, corresponding to a quartic polynomial.
• The leading coefficient is 5, which is positive.

The leading term helps you quickly predict the polynomial's end behavior by applying the Leading Coefficient Test. Because \( 5x^4 \) has a positive coefficient and an even degree, it suggests the graph will rise on both ends.