Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$g(x)=-1.6 x^{5}+4 x^{2}-2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = -1.6x^5 + 4x^2 - 2\) falls to the right and rises to the left.

Step 1: Identify Degree and Leading Coefficient

The given polynomial function is \(g(x) = -1.6x^5 + 4x^2 - 2\). The highest power of \(x\) (i.e., the degree of the polynomial) is 5 and the coefficient of this term (i.e., the leading coefficient) is -1.6.

Step 2: Apply the Leading Coefficient Test

As the degree of the polynomial is odd and the leading coefficient is negative, the sign of the leading coefficient indicates that the graph of the function will fall to the right, and the oddness of the degree indicates the graph will rise to the left.

Step 3: Interpret the Result

The Leading Coefficient Test tells us that the right-hand behavior of the graph is to fall and the left-hand behavior of the graph is to rise.

Key Concepts

Polynomial Function Behavior

The behavior of polynomial functions is an essential aspect of understanding how their graphs behave as input values become very large or very small. This behavior is mainly dictated by the terms with the highest degree of the polynomial. Essentially, for a polynomial function like \( g(x) = -1.6x^5 + 4x^2 - 2 \), it's primarily the term \( -1.6x^5 \) that will influence the behavior of the graph at the extremes.

• As \( x \to \infty \) or \( x \to -\infty \), the higher power terms dominate.
• The behavior is divided into two types: right-hand and left-hand behavior.
• Right-hand behavior describes what happens to the function's value as \( x \to \infty \).
• Left-hand behavior describes the function's value as \( x \to -\infty \).

Understanding this behavior helps us predict whether the graph rises or falls as \( x \) moves outward in either direction. In our example, the graph's falling or rising nature at the ends will be guided by the analysis of its degree and leading coefficient.

Degree of a Polynomial

The degree of a polynomial is defined as the highest power of the variable in the polynomial. It tells us important information about the polynomial's potential behavior.

• The degree influences the possible number of roots the polynomial can have.
• It helps determine the shape and endpoints of the graph.
• Even degree polynomials typically have graphs that rise or fall together on both ends.
• Odd degree polynomials have graphs that have opposing end behaviors (one end up and one end down).

In the case of the polynomial \( g(x) = -1.6x^5 + 4x^2 - 2 \), its degree is 5 because the highest power of \( x \) is 5. Being an odd degree, it informs us that the graph will have opposite behaviors at each end, meaning if one end rises, the other will fall.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It plays a pivotal role in determining the general direction of the graph's ends.

• Positive leading coefficients in odd degree polynomials result in right-end rising and left-end falling.
• Negative leading coefficients in odd degree polynomials result in right-end falling and left-end rising.
• For even degree polynomials, positive leads result in both ends rising, and negatives result in both ends falling.

In our specific example of \( g(x) = -1.6x^5 + 4x^2 - 2 \), the leading coefficient, which is \(-1.6\), tells us that the right-hand side of the graph will fall because it is negative, while the left-hand side will rise due to the odd degree nature.