Problem 36 - 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. \(f(s)=-\frac{7}{8}\left(s^{3}+5 s^{2}-7 s+1\right)\)

Step-by-Step Solution

Answer

Under the Leading Coefficient Test, since the leading coefficient of our polynomial function is negative and the degree is odd, the right end of the graph will point downwards and the left end will point upwards. This is confirmed when the function is graphed.

Step 1: Identify the Leading Coefficient and Degree of the Polynomial

Observe that the given polynomial \(f(s)=-\frac{7}{8}\left(s^{3}+5 s^{2}-7 s+1\right)\) is already in standard form, that is, the terms are in decreasing order of exponents. The leading coefficient is -7/8, and the degree of the polynomial is 3.

Step 2: Apply the Leading Coefficient Test Rule

Based on the Leading Coefficient Test, for the polynomial of odd degree, the sign of the leading coefficient determines whether the ends of the graph go to opposite directions. If the leading coefficient is positive, the right end of the graph will be up and the left end will be down. Conversely if it is negative, the right end will be down and the left end will be up. Here, the leading coefficient is negative - hence the right end of the graph is down and the left end is up.

Step 3: Confirming the Result Using a Graphing Tool

By plotting the polynomial \(f(s)=-\frac{7}{8}\left(s^{3}+5 s^{2}-7 s+1\right)\) using a graphing tool, you can verify that the right end of the graph points downwards and the left end points upwards, thus confirming the results of the Leading Coefficient Test.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions made up of variables and coefficients. They feature one or more terms, where each term is a product of a constant coefficient and a variable raised to a non-negative integer power. For example, in the polynomial function \(f(s) = -\frac{7}{8}(s^{3} + 5s^{2} - 7s + 1)\), the terms are \(-\frac{7}{8}s^3\), \(-\frac{35}{8}s^2\), \(\frac{7}{8}s\), and \(-\frac{7}{8}\).
Basically, polynomials are categorized based on their degree, which is the highest power of the variable present. In our example, the degree is 3 because the term \(s^3\) is the highest power of the variable. Polynomial functions can have various shapes when plotted on a graph, depending on their coefficients and degree. Understanding the basic structure of polynomial functions helps in predicting their overall behavior on the graph.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the polynomial equation on a coordinate plane to understand its shape and behavior. The graph of a polynomial function reveals where it hits the x-axis (its roots) and how it curves in response to the coefficients of different terms.
The leading coefficient and the degree of the polynomial significantly influence the graph's shape, particularly in determining the end behavior. For example, the Leading Coefficient Test is a technique used to predict how the graph behaves as it extends towards positive or negative infinity. In the case of \(f(s) = -\frac{7}{8}(s^{3} + 5s^{2} - 7s + 1)\), the graph can be plotted using graphing tools like graphing calculators or software packages. This visualization assists in verifying the predictions made by applying rules like the Leading Coefficient Test.

Odd Degree Polynomials

Odd degree polynomials are those whose highest exponent of the variable is an odd number. These polynomials, like \(f(s) = -\frac{7}{8}(s^{3} + 5s^{2} - 7s + 1)\), typically exhibit a characteristic behavior: their graphs extend in opposite directions on each side. This means one end of the graph points towards positive infinity while the other points towards negative infinity.
The specific direction of these ends is determined by the sign of the leading coefficient. For instance, if the leading coefficient is positive, the graph ascends to the right and descends to the left. Conversely, if the leading coefficient is negative, as in our example, the graph descends to the right and ascends to the left. Understanding such properties of odd degree polynomials makes predicting and analyzing their graphs easier and aids in solving various mathematical problems related to them.