Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=12 x^{107,499}$$

Step-by-Step Solution

Answer

Rises as \( x \to \infty \) and falls as \( x \to -\infty \).

Step 1: Identify Degree and Leading Coefficient

The given polynomial function is \( P(x) = 12x^{107,499} \). The degree of the polynomial, which is the highest exponent of \( x \), is 107,499. The leading coefficient, which is the coefficient of the term with the highest degree, is 12.

Step 2: Determine End Behavior Based on Degree

For any polynomial, the end behavior depends on the degree being even or odd. Here, 107,499 is an odd number. When a polynomial has an odd degree, the ends of the graph will move in opposite directions.

Step 3: Analyze Leading Coefficient's Effect on End Behavior

Since the leading coefficient is 12, which is positive, the right end (as \( x \to \infty \)) of the graph of \( P(x) \) will rise towards positive infinity. For an odd degree polynomial with a positive leading coefficient, as \( x \to -fty \), the left end of the graph will fall towards negative infinity.

Step 4: Summarize End Behavior

From the analysis, the end behavior of the function \( P(x) \) is: as \( x \to \infty \), \( P(x) \to fty \) (graph rises); and as \( x \to -fty \), \( P(x) \to -fty \) (graph falls).

Key Concepts

Polynomial Degree

The degree of a polynomial is a fundamental characteristic that determines its overall shape and behavior. Specifically, it's the highest power of the variable, denoted as the exponent of the leading term. For the function \( P(x) = 12x^{107,499} \), the degree is 107,499. This value plays a crucial role, especially when predicting the end behavior of the polynomial's graph.

Polynomial degree can be either even or odd, which significantly impacts the directions in which the ends of the graph point. In simple terms:

• If the degree is even, both ends of the polynomial graph will tend to move in the same direction.
• If the degree is odd, like in our example with 107,499, the ends will move in opposite directions.

Thus, understanding the degree helps us predict the behavior without graphing, making it a powerful tool in polynomial analysis.

Leading Coefficient

The leading coefficient of a polynomial is another critical factor when discussing end behavior. It is the coefficient of the term with the highest degree, shaping how drastic or gentle the rise or fall of the graph is as \( x \) approaches infinity or negative infinity.

In the polynomial \( P(x) = 12x^{107,499} \), the leading coefficient is 12, which is positive. Here's how the sign of the leading coefficient affects the end behavior:

• If it's positive, as \( x \to \infty \), the graph will rise to positive infinity.
• Conversely, if it's negative, the same end will fall to negative infinity.

For our polynomial with a positive leading coefficient, it's straightforward to determine that the right side of the graph rises with increasing \( x \). This characteristic becomes particularly useful in tandem with understanding the polynomial's degree.

Odd Degree Polynomial

An odd degree polynomial, such as \( P(x) = 12x^{107,499} \), is defined by its degree being an odd number—in this case, 107,499. This odd nature of the degree heavily influences how the ends of the polynomial graph behave.

When dealing with an odd degree:

• The graph will have one end approaching positive infinity, while the other reaches toward negative infinity.
• For a positive leading coefficient, such as 12, the left end (as \( x \to -\infty \)) will drop to negative infinity, and the right end (as \( x \to \infty \)) will soar to positive infinity.

This dual-direction behavior grants odd degree polynomials their unique characteristic shape and makes them distinct from their even degree counterparts. Understanding this concept allows for not only predicting end behaviors but also grasping the foundational symmetry and directional properties inherent in polynomial graphs.