Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Describe the right-hand and left-hand behavior of the graph of the polynomial function. $$f(s)=-\frac{2}{8}\left(s^{3}+5 s^{2}-7 s+1\right)$$

Step-by-Step Solution

Answer

The graph of the function falls to the left and rises to the right. As s moves left (towards negative infinity), f(s) increases towards positive infinity. As s moves right (towards positive infinity), f(s) decreases towards negative infinity.

Step 1: Simplify the given function

First, simplify the given function \( f(s) = -\frac{2}{8}(s^{3} + 5s^{2} - 7s + 1) \) by multiplying the terms inside the brackets with the factor outside the brackets (-2/8). The simplified version of the function is \( f(s) = -\frac{1}{4}s^{3} - \frac{5}{4}s^{2} + \frac{7}{4}s - \frac{1}{4} \)

Step 2: Determine the leading term of the polynomial

The term of highest degree in a polynomial function is known as the leading term. In this simplified function, the leading term is -1/4 s^{3}.

Step 3: Determine the right-hand and left-hand behavior

The leading term -1/4 s^{3} decides the end behavior of the function. Since the degree of the leading term is odd (3) and its coefficient is negative (-1/4), the graph of the function falls to the left and rises to the right. This means that as s approaches negative infinity (moves left), the function f(s) approaches positive infinity. Conversely, as s approaches positive infinity (moves right), the function f(s) approaches negative infinity.

Key Concepts

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as the input variable approaches positive or negative infinity. It's like observing what happens to the function's graph at its farthest right and left edges. Understanding end behavior helps predict the trend of polynomial graphs without needing a complete plot.

For polynomial functions, the end behavior is primarily determined by the degree of the polynomial and the sign of the leading coefficient.

• If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
• If the degree is odd and the leading coefficient is negative, as in our example, the graph rises to the left and falls to the right.
• If the degree is even and the leading coefficient is positive, the graph rises both to the left and right extremes.
• If the degree is even and the leading coefficient is negative, the graph falls both to the left and right extremes.

Thus, in our function \(-\frac{1}{4}s^3 - \frac{5}{4}s^2 + \frac{7}{4}s - \frac{1}{4}\), the end behavior is determined by the leading term \(-\frac{1}{4}s^3\). This tells us that the graph will rise to the left and fall to the right.

Leading Term and Its Significance

In a polynomial function, the leading term is the term with the highest degree. The leading term can give us direct clues about the polynomial's end behavior and significantly influences the function's overall shape. Recognizing this term is crucial, especially when you want to understand or sketch a polynomial function quickly.

Consider our function \(f(s) = -\frac{1}{4}s^3 - \frac{5}{4}s^2 + \frac{7}{4}s - \frac{1}{4}\). Here, the leading term is \(-\frac{1}{4}s^3\). The presence of the exponent 3 on the variable \(s\) signifies that this is a cubic polynomial, with the behavior characteristics associated with odd-degree terms. Additionally, the negative coefficient \(-\frac{1}{4}\) suggests that the end behavior will deviate from usual upward trends, causing the graph to switch behavior at both ends (rising left, falling right).

Understanding the leading term helps decode much about a polynomial's behavior even before diving into more complex analysis or plotting.

Degree of a Polynomial Explained

The degree of a polynomial is the highest power of the variable in the expression. It plays a pivotal role in shaping the graph of the polynomial function and speaks volumes about its nature.

For our polynomial function \(f(s) = -\frac{1}{4}s^3 - \frac{5}{4}s^2 + \frac{7}{4}s - \frac{1}{4}\), the degree is 3. This defines it as a cubic polynomial, dictating that the function will have more dynamic shifts and turns compared to simpler linear or quadratic polynomials.

The degree not only influences the number of potential zero crossings or turning points but also the overall end behavior as discussed earlier.

• Polynomials with even degrees tend to have similar end behavior on both sides of the graph, either rising or falling.
• Polynomials with odd degrees, like ours, have opposite ends trending differently.

The degree is an essential factor for mathematicians and students alike when unlocking the behavior and character of polynomial functions.