Problem 12 - A Graphical Approach to College Algebra

Question

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\pi x^{7}-x^{5}+x-1$$

Step-by-Step Solution

Answer

As \( x \to -\infty \), \( P(x) \to -\infty \) and as \( x \to +\infty \), \( P(x) \to +\infty \).

Step 1: Identify the Degree and Leading Coefficient

The polynomial function given is \( P(x) = \pi x^{7} - x^{5} + x - 1 \). The highest degree term is \( \pi x^7 \), making the degree 7. The coefficient of this term is \( \pi \), which is a positive constant.

Step 2: Determine End Behavior Based on Degree

Since the degree of the polynomial is 7, which is odd, the end behavior of the polynomial will mirror the behavior of \( x^7 \). Odd-degree polynomials have opposite behaviors on each end.

Step 3: Analyze the Positive Leading Coefficient Impact

The leading coefficient (\( \pi \)) is positive. For odd-degree polynomials, this implies that as \( x \to -\infty \) (left end), \( P(x) \to -\infty \). As \( x \to +\infty \) (right end), \( P(x) \to +\infty \).

Step 4: Write the End Behavior Description

Using the information from the previous steps, we can describe the end behavior of the polynomial \( P(x) = \pi x^7 - x^5 + x - 1 \). As \( x \to -\infty \), \( P(x) \to -\infty \) and as \( x \to +\infty \), \( P(x) \to +\infty \).

Key Concepts

Degree of Polynomial

When discussing polynomials, the term "degree" refers to the highest power of the variable present in the polynomial expression. For any polynomial, the degree indicates the maximum number of solutions or roots the equation can have. It essentially tells us about the polynomial's broad structure and how intricate its graph might become.

To identify the degree of a polynomial, search for the term with the highest exponent. In our given polynomial \( P(x) = \pi x^7 - x^5 + x - 1 \), the term \( \pi x^7 \) has the highest exponent, making the degree 7. This indicates that the polynomial can potentially intersect the x-axis up to 7 times.

Remember, higher degree polynomials generally result in more complex graphs, with perhaps more turning points and changes in direction. The degree of a polynomial is a fundamental aspect in predicting how the graph behaves, especially far from the origin.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of a graph.

In our polynomial \( P(x) = \pi x^7 - x^5 + x - 1 \), the leading term is \( \pi x^7 \) and the leading coefficient is \( \pi \), which is a positive number. This coefficient affects how the polynomial behaves as the variable approaches extremely large positive or negative values.

• If the leading coefficient is positive, for an odd-degree polynomial, the graph will rise to the right and fall to the left.
• If the leading coefficient is negative, the graph will fall to the right and rise to the left.

This information is critical when sketching the graph since it outlines how the polynomial will behave at its edges or extremities.

Odd-Degree Polynomial

An odd-degree polynomial is one where the highest power of the variable is an odd number. Polynomials with odd degrees have distinctive end behaviors because they involve terms that cause the graph to point in opposite directions at infinity.

For the polynomial \( P(x) = \pi x^7 - x^5 + x - 1 \), the degree is 7, which is odd. This means the function's graph will behave differently at each end. Specifically:

• For \( x \to -\infty \), the graph heads in one direction (normally downward if the leading coefficient is positive).
• For \( x \to +\infty \), the graph goes in the opposite direction (upward if the leading coefficient is positive).

Thus, odd-degree polynomials sharply demonstrate contrasting behaviors at either end of their graph. Understanding whether a polynomial is of odd degree is essential for predicting its end behavior and sketching its graph accurately.