Problem 32 - Precalculus : Building Concepts and Connections

Question

Determine the end behavior of the function. $$f(s)=-\frac{3}{4} s^{4}+8 s^{2}-3 s-16$$

Step-by-Step Solution

Answer

The end behavior of the function \(f(s)\) is: as \(s \rightarrow -\infty\), \(f(s)\rightarrow +\infty\) and as \(s \rightarrow +\infty\), \(f(s)\rightarrow -\infty\).

Step 1: Identify the leading term of the function

In the function \(f(s)=-\frac{3}{4} s^{4}+8 s^{2}-3 s-16\), the leading term is \(-\frac{3}{4} s^{4}\). The leading term of a function is the term with the highest degree (power of \(s\)). Here, the term with the highest degree is \(s^{4}\) and its coefficient is \(-\frac{3}{4}\).

Step 2: Determine the degree and sign of the leading coefficient

For the term \(-\frac{3}{4} s^{4}\), the degree of \(s\) is \(4\) which is an even number. The coefficient of this term is \(-\frac{3}{4}\), which is negative.

Step 3: Determine the end behavior

Here, we have a even degree and a negative leading coefficient. For a polynomial where the degree is even and the leading coefficient is negative, as \(s \rightarrow -\infty\) (when \(s\) approaches negative infinity), \(f(s) \rightarrow +\infty\) (the function approaches positive infinity). And, as \(s \rightarrow +\infty\) (when \(s\) approaches positive infinity), \(f(s) \rightarrow -\infty\) (the function approaches negative infinity). This means our function will rise to the left and fall to the right.

Key Concepts

Polynomial Functions

Polynomial functions are fundamental in mathematics and are formed by adding together terms of the form \(a_n x^n\), where \(a_n\) is a coefficient and \(n\) is a non-negative integer known as the degree of the term. A general polynomial can be written as:

• \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\)

Each term in a polynomial can have different degrees, but there is always one with the highest degree, known as the leading term. Polynomials can be classified based on their degrees:

• Linear (degree 1) - e.g., \(f(x) = 2x + 1\)
• Quadratic (degree 2) - e.g., \(f(x) = x^2 - 3x + 2\)
• Cubic (degree 3) - e.g., \(f(x) = x^3 + x^2 - x\)

Understanding polynomial functions includes knowing how these terms interact with each other, particularly how they influence the behavior and shape of the graph. The combination of all these terms defines the unique shape and properties of a polynomial graph.

Leading Coefficients

The leading coefficient of a polynomial is the coefficient of the leading term, which is the term with the highest degree. In our example \(f(s) = -\frac{3}{4}s^4 + 8s^2 - 3s - 16\), the leading term is \(-\frac{3}{4}s^4\) and thus, the leading coefficient is \(-\frac{3}{4}\). The leading coefficient helps determine the long-term behavior of the polynomial graph, or its "end behavior." This means, as the input value \(s\) moves towards positive or negative infinity, the leading coefficient in combination with the polynomial's degree largely dictates how the function behaves. Key points about leading coefficients:

• If the leading coefficient is positive, the ends of the polynomial function will generally go upward.
• If it is negative, the ends will point downward.

This interaction between the leading coefficient and the degree of the polynomial is crucial for predicting the end behavior of any polynomial function.

Degree of a Polynomial

The degree of a polynomial is the highest degree of any term with a nonzero coefficient in the polynomial. It provides valuable information about the polynomial, such as the number of roots it can have and the behavior of the graph. For example, in our polynomial \(f(s) = -\frac{3}{4}s^4 + 8s^2 - 3s - 16\), the degree is 4 because the highest power of \(s\) is 4. The degree tells us several things about the polynomial:

• An even degree means both ends of the graph will tend in the same direction.
• An odd degree means the ends will aim in opposite directions.

In cases where the degree is even and the leading coefficient is negative, as demonstrated in this exercise, the polynomial's ends will rise on the left and fall on the right. Conversely, if the leading coefficient were positive, both ends would rise. Understanding these properties helps in sketching and analyzing polynomial functions.