1.7 A Rational Functions And End Behavior Answer Key

1.7 Rational Functions and End Behavior: A Comprehensive Guide

Understanding the end behavior of rational functions is crucial for mastering precalculus and calculus. This comprehensive guide will delve into the intricacies of rational functions, exploring their characteristics, analyzing their end behavior, and providing you with the tools to confidently tackle any related problem. We'll move beyond simple answers and equip you with a robust understanding of the underlying principles.

What are Rational Functions?

A rational function is defined as the ratio of two polynomial functions, f(x) = p(x) / q(x), where p(x) and q(x) are polynomials, and q(x) is not the zero polynomial (meaning it can't be identically zero for all x). The domain of a rational function is all real numbers except for the values of x that make the denominator, q(x), equal to zero. These values are called vertical asymptotes.

Key Characteristics of Rational Functions:

• Vertical Asymptotes: These are vertical lines (x = a) where the function approaches positive or negative infinity. They occur at values of x that make the denominator zero but do not cancel out with a factor in the numerator.

• Horizontal Asymptotes: These are horizontal lines (y = b) that the function approaches as x goes to positive or negative infinity. They describe the end behavior of the function.

• Oblique (Slant) Asymptotes: These are slanted lines that the function approaches as x goes to positive or negative infinity. They occur when the degree of the numerator is exactly one greater than the degree of the denominator.

• Holes (Removable Discontinuities): These occur when a factor in the numerator and denominator cancel out. The function is undefined at the x-value that makes the cancelled factor zero, but there's a "hole" in the graph at that point.

• x-intercepts (Roots or Zeros): These are the points where the graph intersects the x-axis (y = 0). They occur when the numerator is equal to zero and the denominator is not.

• y-intercept: This is the point where the graph intersects the y-axis (x = 0). It's found by evaluating f(0), provided that f(0) is defined.

Analyzing End Behavior: Horizontal Asymptotes

The end behavior of a rational function describes what happens to the function's values as x approaches positive or negative infinity. This is primarily determined by the degrees of the numerator and denominator polynomials.

Determining Horizontal Asymptotes:

There are three cases to consider when determining the horizontal asymptote:

Case 1: Degree of Numerator < Degree of Denominator

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0 (the x-axis). As x approaches infinity, the denominator grows much faster than the numerator, causing the fraction to approach zero.

Example: f(x) = (2x + 1) / (x² - 4) Here, the degree of the numerator (1) is less than the degree of the denominator (2), so the horizontal asymptote is y = 0.

Case 2: Degree of Numerator = Degree of Denominator

If the degrees of the numerator and denominator are equal, the horizontal asymptote is y = a/b, where 'a' is the leading coefficient of the numerator and 'b' is the leading coefficient of the denominator.

Example: f(x) = (3x² + 2x - 1) / (x² + 5) Here, the degrees are equal (both 2). The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = 3/1 = 3.

Case 3: Degree of Numerator > Degree of Denominator

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, the function may have an oblique (slant) asymptote or approach positive or negative infinity as x approaches infinity.

Example: f(x) = (x³ + 2x) / (x² - 1) The degree of the numerator (3) is greater than the degree of the denominator (2). This function has no horizontal asymptote. To find the slant asymptote, perform polynomial long division.

Analyzing End Behavior: Oblique Asymptotes

As mentioned, oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find the equation of the oblique asymptote, perform polynomial long division. The quotient (excluding the remainder) represents the equation of the slant asymptote.

Example: Let's consider f(x) = (x³ + 2x) / (x² - 1). Performing long division:

       x
x² - 1 | x³ + 0x² + 2x + 0
       - (x³     - x)
       -------------
             x + 0

The quotient is x. Therefore, the oblique asymptote is y = x. As x approaches infinity, the function's values approach the line y = x.

Solving Problems: A Step-by-Step Approach

Let's work through some examples to solidify our understanding:

Problem 1: Find the horizontal and vertical asymptotes of f(x) = (x - 2) / (x² - 4).

Solution:

1. Vertical Asymptotes: Factor the denominator: x² - 4 = (x - 2)(x + 2). The denominator is zero when x = 2 or x = -2. However, the factor (x - 2) cancels out with the numerator. This means there's a hole at x = 2. The only vertical asymptote is x = -2.

2. Horizontal Asymptote: The degree of the numerator (1) is less than the degree of the denominator (2). Therefore, the horizontal asymptote is y = 0.

Problem 2: Find the horizontal and vertical asymptotes and any holes for g(x) = (x² + x - 6) / (x² - 9).

Solution:

1. Factor: g(x) = (x + 3)(x - 2) / (x + 3)(x - 3)

2. Simplify: The (x + 3) term cancels, leaving g(x) = (x - 2) / (x - 3), with a hole at x = -3.

3. Vertical Asymptote: The denominator is zero when x = 3. This is the vertical asymptote.

4. Horizontal Asymptote: The degrees of the numerator and denominator are equal (both 1). The leading coefficients are both 1, so the horizontal asymptote is y = 1/1 = 1.

Problem 3: Determine the end behavior of h(x) = (2x³ + 5x) / (x² - 1).

Solution:

The degree of the numerator (3) is greater than the degree of the denominator (2). Therefore, there's no horizontal asymptote. To find the oblique asymptote, we perform polynomial long division:

      2x
x² - 1 | 2x³ + 0x² + 5x
      - (2x³     -2x)
      -------------
              7x

The oblique asymptote is y = 2x. As x approaches positive infinity, h(x) approaches positive infinity along the line y = 2x. As x approaches negative infinity, h(x) approaches negative infinity along the line y = 2x.

Advanced Considerations and Applications

The concepts discussed here form the foundation for more advanced topics in calculus, such as limits, derivatives, and integrals involving rational functions. Understanding end behavior is essential for sketching accurate graphs of rational functions, analyzing their behavior near asymptotes, and solving related optimization problems. Furthermore, rational functions are ubiquitous in modeling various real-world phenomena, including population growth, decay processes, and electrical circuits.

Conclusion

Mastering the end behavior of rational functions is vital for a thorough understanding of their properties. By carefully analyzing the degrees of the numerator and denominator polynomials, you can accurately determine the presence and equations of horizontal and oblique asymptotes, enabling you to sketch graphs and solve problems effectively. Remember to always check for holes (removable discontinuities) by factoring and canceling common factors in the numerator and denominator. Practice is key; the more problems you solve, the more comfortable and proficient you'll become in working with these essential functions. This detailed guide equips you with the necessary knowledge and strategies to tackle any challenge related to rational functions and their end behavior confidently.