Problem 20 - College Algebra

Question

Determine the end behavior of the functions. $$f(x)=-x^{9}$$

Step-by-Step Solution

Answer

As \( x \to \text{∞} \), \( f(x) \to \text{-∞} \); as \( x \to \text{-∞} \), \( f(x) \to \text{∞} \).

Step 1: Identify the Leading Term

The leading term of the function given is \( -x^9 \), which is the term with the highest exponent.

Step 2: Determine the Degree and Sign

The degree of the polynomial is \( 9 \) because of the term \( x^9 \). It's an odd degree, and the leading coefficient is negative.

Step 3: Analyze End Behavior for Odd Degree and Negative Leading Coefficient

For functions of the form \( -x^n \) where \( n \) is odd, as \( x \rightarrow \, ext{∞} \), \( f(x) \rightarrow \, - ext{∞} \), and as \( x \rightarrow \, - ext{∞} \), \( f(x) \rightarrow \, ext{∞} \). This happens because the negative leading coefficient flips the standard end behavior of odd degree functions.

Key Concepts

Leading Term

When working with polynomial functions, the leading term is crucially important. It is the term with the highest degree of the variable, which decisively influences the polynomial's end behavior. In our example, the leading term is \( -x^9 \). This means the highest power of \( x \) is 9, and it helps us understand the function's growth pattern as \( x \) becomes very large or very small.
The leading term's degree and coefficient dictate how steeply the function rises or falls. Understanding the leading term first allows us to make predictions about the polynomial's end behavior without graphing it. This leading indicator sets the stage for analyzing other aspects like degree and sign. See it as the polynomial's loudest voice, commanding the function's overall direction as \( x \) stretches towards infinity.

Odd Degree Polynomial

Polynomials come in various degrees, reflecting the highest exponent's value within the expression. Here, we are dealing with an odd degree polynomial, specifically of degree 9.
Odd degree polynomials are unique because their graphs generally have opposite end behaviors. In other words, as \( x \) moves towards positive infinity, \( f(x) \) will head in one direction, whereas as \( x \) moves towards negative infinity, \( f(x) \) heads in the other direction.
The degree of a polynomial not only tells you about the end behavior but also about the graph's shape. With an odd degree, the function will have one tail going up to infinity and the other tail going down to negative infinity, unless affected by the sign of the leading coefficient.

Negative Leading Coefficient

The sign of the leading coefficient, which is negative in this case, brings about a significant influence on the polynomial's end behavior. When the leading coefficient is negative, it serves to reflect the polynomial's graph across the x-axis compared to what would happen if the coefficient were positive.
For our example \( -x^9 \):

• As \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \)
• As \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \)

This inversion means that while an odd degree polynomial with a positive leading coefficient would rise to \( \infty \) on one end and fall to \(-\infty \) on the other, here it's entirely the opposite. The negative sign implies the tails of the polynomial switch orientations, which is a key aspect to grasp for correctly predicting the function's end behavior.