Problem 22 - Algebra and Trigonometry

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. \(f(x)=11 x^{4}-6 x^{2}+x+3\)

Step-by-Step Solution

Answer

The end behavior of the graph of the given polynomial function is that it rises as x approaches both positive and negative infinity, i.e., it rises to the right and left.

Step 1: Identify the Leading Term

The leading term of the polynomial \(f(x) = 11x^{4}-6x^{2}+x+3\) is the term with the highest power of x, which is \(11x^{4}\).

Step 2: Identify the Leading Coefficient and Degree of Polynomial

The leading coefficient is the numerical coefficient of the leading term. So, the leading coefficient for this polynomial is 11. The degree of the polynomial is the exponent of the leading term, which is 4 in this case.

Step 3: Apply the Leading Coefficient Test

The degree of the polynomial function is even, and the leading coefficient is positive. According to the Leading Coefficient Test, when the degree is even and the leading coefficient is positive, the end behavior of the graph will rise to the right and rise to the left.

Key Concepts

End Behavior

In mathematics, understanding the end behavior of a polynomial function's graph is crucial. The end behavior describes what happens to the graph of a function as \(x\) approaches positive or negative infinity. Essentially, it's the 'tail' end of the graph.

• If the degree of the polynomial is even and the leading coefficient is positive, the graph's ends will both head upwards.
• If the degree is even and the leading coefficient is negative, the ends of the graph will both head downwards.
• If the degree is odd and the leading coefficient is positive, the graph will fall to the left and rise to the right.
• If the degree is odd and the leading coefficient is negative, the graph will rise to the left and fall to the right.

The end behavior helps us to sketch and understand the graph quickly, making it easier to predict how the polynomial function behaves as values of \(x\) grow large or small.

Polynomial Function

A polynomial function is a mathematical expression consisting of variables (usually represented as \(x\)), coefficients, and exponents, combined using addition, subtraction, and multiplication.
Polynomials are classified based on their degree, which is the highest power of the variable present in the function. They are central in algebra due to their straightforward operations and critical role in modeling real-life situations.
For example, the given polynomial function \(f(x) = 11x^{4} - 6x^{2} + x + 3\) is expressed with four terms. Each term contributes differently to the polynomial's overall shape and behavior.

Leading Term

In a polynomial function, the leading term is the term with the highest power of the variable, \(x\). It plays a dominant role in determining the end behavior of the polynomial. By identifying the leading term, we can simplify the process of applying the Leading Coefficient Test.
For our polynomial \(f(x) = 11x^{4} - 6x^{2} + x + 3\), the leading term is \(11x^{4}\). This term indicates that the behavior of the polynomial at large values of \(x\) is primarily influenced by this \(x^4\) term rather than the smaller degree terms.

Degree of Polynomial

The degree of a polynomial is the highest exponent of the variable \(x\) within the polynomial expression. It is crucial in determining the polynomial’s shape and its end behavior.

• Polynomials of even degree tend to have graphs that either rise or fall in both directions.
• Polynomials of odd degree typically have graphs that fall on one side and rise on the opposite side of the graph.

For \(f(x) = 11x^{4} - 6x^{2} + x + 3\), the degree is 4, indicating that it is a quartic polynomial. Being an even degree, this means the graph will exhibit symmetric end behavior.