Problem 6 - Calculus Early Transcendentals

Question

Describe the possible end behavior of a polynomial.

Step-by-Step Solution

Answer

Answer: The end behavior of a polynomial function depends on its degree and leading coefficient in the following ways: 1. If the polynomial has an even degree and a positive leading coefficient, both ends of the graph will approach positive infinity. 2. If the polynomial has an even degree and a negative leading coefficient, both ends of the graph will approach negative infinity. 3. If the polynomial has an odd degree and a positive leading coefficient, one end of the graph will approach negative infinity, while the other end will approach positive infinity. 4. If the polynomial has an odd degree and a negative leading coefficient, one end of the graph will approach positive infinity, while the other end will approach negative infinity.

Step 1: Understand the degree and leading coefficient

A polynomial function can be written in the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n\) is the leading coefficient and \(n\) is the degree of the polynomial. The degree is the highest power of the variable \(x\) in the polynomial, and the leading coefficient is the coefficient of the term with the highest degree.

Step 2: Identify the properties of even and odd degree polynomials

There are two fundamental properties of even and odd degree polynomials that affect end behavior: 1. If the polynomial has an even degree, then its end behavior will be the same in both directions. That is, if one end of the graph goes to positive infinity, then the other end will also go to positive infinity; likewise, if one end goes to negative infinity, then the other end will also go to negative infinity. 2. If the polynomial has an odd degree, then its end behavior will be different in both directions. If one end of the graph goes to positive infinity, then the other end will go to negative infinity, and vice versa.

Step 3: Describe the end behavior based on the degree and leading coefficient

Now that we understand how the degree and leading coefficient affect the end behavior, we can describe it for the 4 possible scenarios: 1. Even degree and positive leading coefficient: As \(x\) approaches negative infinity, \(P(x)\) will approach positive infinity; as \(x\) approaches positive infinity, \(P(x)\) will also approach positive infinity. 2. Even degree and negative leading coefficient: As \(x\) approaches negative infinity, \(P(x)\) will approach negative infinity; as \(x\) approaches positive infinity, \(P(x)\) will also approach negative infinity. 3. Odd degree and positive leading coefficient: As \(x\) approaches negative infinity, \(P(x)\) will approach negative infinity; as \(x\) approaches positive infinity, \(P(x)\) will approach positive infinity. 4. Odd degree and negative leading coefficient: As \(x\) approaches negative infinity, \(P(x)\) will approach positive infinity; as \(x\) approaches positive infinity, \(P(x)\) will approach negative infinity.

Step 4: Illustrate end behavior using examples and graphical representations

To better understand the end behavior of each scenario, we can use examples: 1. Even degree and positive leading coefficient: The polynomial \(P(x) = x^2\), since \(n=2\) (even degree) and \(a_n=1\) (positive leading coefficient). The end behavior is: - As \(x\) approaches negative infinity, \(P(x)\) approaches positive infinity. - As \(x\) approaches positive infinity, \(P(x)\) approaches positive infinity. 2. Even degree and negative leading coefficient: The polynomial \(P(x) = -x^4\), since \(n=4\) (even degree) and \(a_n=-1\) (negative leading coefficient). The end behavior is: - As \(x\) approaches negative infinity, \(P(x)\) approaches negative infinity. - As \(x\) approaches positive infinity, \(P(x)\) approaches negative infinity. 3. Odd degree and positive leading coefficient: The polynomial \(P(x) = x^3\), since \(n=3\) (odd degree) and \(a_n=1\) (positive leading coefficient). The end behavior is: - As \(x\) approaches negative infinity, \(P(x)\) approaches negative infinity. - As \(x\) approaches positive infinity, \(P(x)\) approaches positive infinity. 4. Odd degree and negative leading coefficient: The polynomial \(P(x) = -x^5\), since \(n=5\) (odd degree) and \(a_n=-1\) (negative leading coefficient). The end behavior is: - As \(x\) approaches negative infinity, \(P(x)\) approaches positive infinity. - As \(x\) approaches positive infinity, \(P(x)\) approaches negative infinity.