Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(h(x)=1-x^{6}\)

Step-by-Step Solution

Answer

The right-hand and left-hand behavior of the graph of the function \(h(x) = 1-x^{6}\) both fall according to the Leading Coefficient Test. A graphing tool confirms this behavior.

Step 1: Identify Degree and Leading Coefficient

The first step is to identify the degree and the leading coefficient of the polynomial. The degree of the polynomial function \(h(x) = 1-x^{6}\) is 6, and the leading coefficient (the coefficient of the term with the highest power) is -1.

Step 2: Apply Leading Coefficient Test

The Leading Coefficient Test gives us the behavior of the graph to the extreme left and right. The degree of our polynomial is even and the leading coefficient is negative. For an even degree: -If the leading coefficient is positive, the graph rises to the right and rises to the left. -If the leading coefficient is negative, the graph falls to the right and falls to the left. So, according to the test, the graph of \(h(x) = 1-x^{6}\) falls to the right and falls to the left.

Step 3: Verify Results with Graphing Utility

Plot the function \(h(x) = 1-x^{6}\) using a graphing utility. As observed, the graph goes down or 'falls' as x tends to positive infinity (to the right) and also as x tends to negative infinity (to the left). This confirms the results obtained from the Leading Coefficient Test.

Key Concepts

Polynomial Function Behavior

Understanding the behavior of polynomial functions is crucial in the study of calculus and algebra. Polynomial function behavior primarily refers to how the graph of a polynomial function behaves as the input values, or 'x', become very large in either the positive or negative direction. In the context of the leading coefficient test for the polynomial function h(x) = 1 - x^6, we learned that the degree (highest power of 'x') and the leading coefficient (the coefficient of the term with the highest power) are instrumental in predicting this behavior.

The test reveals that the end behavior of the polynomial function does indeed depend on these two elements. For example, in our exercise, because the degree is even and the leading coefficient is negative, the behavior at both ends falls off, which means the graph of the function points downwards in both directions as 'x' moves away from zero. This manifestation is a fundamental characteristic of even-degree polynomials with a negative leading coefficient and it allows us to sketch an approximate graph of this polynomial function without precise calculations or a graphing utility.

Graphing Polynomials

Graphing polynomials involves plotting the function on a coordinate system to visualize its shape and important features, such as x-intercepts, y-intercepts, turning points, and end behavior. By graphing the given polynomial h(x) = 1 - x^6, we are able to verify the predictions made by the Leading Coefficient Test.

A graphing utility helps in crafting the precise curve of the polynomial, taking into account all powers of 'x'. Such tools accurately demonstrate the falling behavior on both ends that was theorized, aligning with our initial conclusion about the graph's descent to infinity in both directions. Through graphing, students can confirm their understanding of the polynomial's characteristics and gain a more intuitive grasp of function behaviors.

Degree of a Polynomial

The degree of a polynomial is defined by the highest power of 'x' in its expression. It has a profound impact on many aspects of the polynomial's graph, including its shape, the number of turning points it may have, and its end behavior. An even-degree polynomial, such as the sixth-degree polynomial h(x) = 1 - x^6 from our exercise, often has a mirror-symmetry about the y-axis and the same end behaviors on both sides of the graph.

Conversely, polynomials with an odd degree have opposite end behaviors—with the graph extending to infinity in one direction and negative infinity in the other. The degree of a polynomial along with the leading coefficient are foundational concepts in algebra that facilitate the prediction of a polynomial function's global behavior without requiring detailed graphing for every single point.