Problem 26 - College Algebra

Question

In Exercises \(21-26,\) 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 function \(f(x) = -11x^4 - 6x^2 + x + 3\) is as follows: as x approaches positive infinity, \(f(x)\) approaches negative infinity; as x approaches negative infinity, \(f(x)\) approaches negative infinity.

Step 1: Identify the Leading Term

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

Step 2: Check Leading Coefficient and Degree

The leading coefficient is -11 and the degree of the polynomial is 4. Since 4 is an even integer, regardless of the coefficient's sign, as x approaches both positive and negative infinity, y will both go in the same direction. The sign of the leading coefficient will determine whether they both go towards positive or negative infinity. As the leading coefficient is negative, pointing downwards, as x approaches both positive and negative infinity, y will decrease.

Step 3: Conclude End Behavior

According to the Leading Coefficient Test, because the degree of the function \(f(x)\) is even and the leading coefficient is negative, the end behavior of the function is: as x approaches positive infinity, \(f(x)\) approaches negative infinity, and as x approaches negative infinity, \(f(x)\) approaches negative infinity.

Key Concepts

End Behavior

In mathematics, understanding the end behavior of a polynomial function is key to predicting how the graph behaves as x moves towards positive or negative infinity. The end behavior describes the trends exhibited by the values of the function at the far ends of its domain. For any polynomial function, the leading term (the term with the highest degree) primarily determines this trend.

To determine the end behavior, observe both the degree of the polynomial and the sign of its leading coefficient. If the degree is even, the directions of the ends will match (both will up or both will down). Conversely, if the degree is odd, the directions will differ (one will up, and one will down). Additionally, a positive leading coefficient indicates the graph will rise to positive infinity, while a negative one suggests it will fall to negative infinity.

Polynomial Function

A polynomial function is a mathematical expression consisting of variables raised to whole number powers and coefficients. They are versatile and can represent a wide range of curves and systems. A polynomial function takes the form of:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\]

Where each \(a_n\) is a coefficient. The functions can be classified by their degree (the highest power of x present). For instance, a polynomial of degree 4, like \(f(x) = -11x^4-6x^2+x+3\), is called a quartic polynomial.

• Polynomial functions with degree 0 are constant functions.
• Linear functions have a degree of 1.
• Quadratic functions possess a degree of 2.
• Cubic functions are degree 3 polynomials.

Leading Term

The leading term of a polynomial is crucial for analyzing its graph, especially when it comes to determining end behavior. In any polynomial, the leading term is the term with the highest degree. It usually appears first when the polynomial is written in standard form.

For example, in the polynomial \(f(x) = -11x^4 - 6x^2 + x + 3\), the leading term is \(-11x^4\).

• The coefficient of the leading term is known as the leading coefficient, which greatly influences the steepness and direction of the graph.
• The leading term reveals both the degree and direction of the polynomial's end behavior because it dominates the polynomial's behavior for extreme values of x.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable (typically x) in the expression. It indicates the number of roots or solutions a polynomial function can have, and importantly, plays a significant role in determining the end behavior.

For the polynomial \(f(x) = -11x^4 - 6x^2 + x + 3\), the degree is 4. This tells us it's a quartic polynomial, indicating:

• An even degree means the ends of the graph will either both rise or both fall.
• In contrast, an odd degree would suggest one end of the graph rises while the other falls.
• The degree, combined with the leading coefficient, informs the overall shape and direction of the polynomial's graph.