Worksheet A Topic 1.7 Rational Functions And End Behavior

Worksheet: Topic 1.7 Rational Functions and End Behavior

Understanding rational functions and their end behavior is crucial for success in algebra and beyond. This comprehensive guide will equip you with the knowledge and skills to master this topic. We'll cover key concepts, provide step-by-step examples, and offer practice problems to solidify your understanding. Let's dive in!

What are Rational Functions?

A rational function is defined as the ratio of two polynomial functions, where the denominator is not the zero polynomial. In other words, it's a function that can be expressed in the form:

f(x) = p(x) / q(x)

where p(x) and q(x) are polynomials, and q(x) ≠ 0.

Understanding the characteristics of these polynomials – their degree, leading coefficients, and roots – is key to analyzing the behavior of the rational function.

Key Features of Rational Functions

Several key features distinguish rational functions from other function types:

• Asymptotes: These are lines that the graph of the function approaches but never touches. There are three main types:

  • Vertical Asymptotes: Occur where the denominator, q(x), is equal to zero and the numerator, p(x), is not zero. These represent values of x where the function is undefined.
  • Horizontal Asymptotes: Describe the behavior of the function as x approaches positive or negative infinity. Their existence and location depend on the degrees of the numerator and denominator polynomials.
  • Oblique (Slant) Asymptotes: Occur when the degree of the numerator is exactly one greater than the degree of the denominator. These are slanted lines that the function approaches as x goes to positive or negative infinity.

• Holes (Removable Discontinuities): These occur when both the numerator and denominator share a common factor that can be cancelled out. The function is undefined at the x-value that makes the cancelled factor zero, but the graph has a "hole" at that point instead of a vertical asymptote.

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

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

End Behavior of Rational Functions

The end behavior of a function describes how the function behaves as x approaches positive infinity (+∞) and negative infinity (−∞). For rational functions, the end behavior is primarily determined by the degrees of the numerator and denominator polynomials and their leading coefficients.

Determining Horizontal Asymptotes

The rules for determining horizontal asymptotes are as follows:

1. Degree of Numerator < Degree of Denominator: The horizontal asymptote is y = 0.

2. Degree of Numerator = Degree of Denominator: The horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).

3. Degree of Numerator > Degree of Denominator: There is no horizontal asymptote. Instead, there will be an oblique (slant) asymptote if the degree of the numerator is exactly one greater than the degree of the denominator. If the degree of the numerator is more than one greater than the degree of the denominator, the function will have no horizontal or oblique asymptote and will increase or decrease without bound as x approaches infinity.

Step-by-Step Examples

Let's work through some examples to illustrate these concepts:

Example 1: f(x) = (2x + 1) / (x - 3)

• Vertical Asymptote: The denominator is zero when x = 3. Therefore, there is a vertical asymptote at x = 3.
• Horizontal Asymptote: The degree of the numerator and denominator are both 1. The horizontal asymptote is y = 2/1 = 2.
• x-intercept: The numerator is zero when 2x + 1 = 0, which means x = -1/2. Therefore, the x-intercept is (-1/2, 0).
• y-intercept: When x = 0, f(0) = (2(0) + 1) / (0 - 3) = -1/3. Therefore, the y-intercept is (0, -1/3).

Example 2: f(x) = (x² + 2x - 3) / (x - 1)

Notice that (x² + 2x - 3) can be factored as (x + 3)(x - 1). Thus:

f(x) = [(x + 3)(x - 1)] / (x - 1) = x + 3, provided x ≠ 1.

• Vertical Asymptote: There is no vertical asymptote because the (x - 1) factor cancels out.
• Horizontal Asymptote: There is no horizontal asymptote because there is a slant asymptote.
• Hole: There is a hole at x = 1. To find the y-coordinate of the hole, substitute x = 1 into the simplified form: y = 1 + 3 = 4. The hole is at (1, 4).
• x-intercept: The simplified equation is y = x + 3; therefore the x-intercept is (-3,0)
• y-intercept: Substitute x = 0: y = 0 + 3 = 3. The y-intercept is (0, 3).

Example 3: f(x) = (x³ + 1) / (x² - 4)

• Vertical Asymptotes: The denominator is zero when x = 2 and x = -2. Therefore, there are vertical asymptotes at x = 2 and x = -2.
• Horizontal Asymptote: There is no horizontal asymptote because the degree of the numerator (3) is greater than the degree of the denominator (2). There is also no oblique asymptote.
• x-intercept: The numerator is zero when x³ + 1 = 0, which means x = -1. Therefore, the x-intercept is (-1, 0).
• y-intercept: When x = 0, f(0) = 1/(-4) = -1/4. The y-intercept is (0, -1/4).

Practice Problems

Here are some practice problems to test your understanding:

1. Find the vertical and horizontal asymptotes, x-intercept, and y-intercept of f(x) = (3x - 6) / (x² - 4).

2. Sketch the graph of f(x) = (x² + x - 6) / (x + 3). Identify any holes or asymptotes.

3. Analyze the end behavior of f(x) = (2x⁴ - 5x² + 1) / (x³ + 2x).

4. Find all asymptotes and intercepts for f(x) = (x²+3x+2)/(x²-1).

5. Determine the vertical asymptotes and the horizontal asymptotes of the function: f(x) = (4x² + 3)/(2x² - 8)

Further Exploration

This worksheet provides a solid foundation in understanding rational functions and their end behavior. To deepen your knowledge, you can explore topics such as:

• Graphing Rational Functions: Use the information about asymptotes, intercepts, and end behavior to accurately sketch the graph of a rational function.

• Solving Rational Inequalities: Apply your understanding of rational functions to solve inequalities involving rational expressions.

• Applications of Rational Functions: Explore real-world applications of rational functions in various fields like physics, engineering, and economics.

By mastering the concepts outlined in this worksheet and practicing regularly, you'll build a strong understanding of rational functions and their behavior, opening doors to more advanced mathematical concepts. Remember to break down complex problems into smaller, manageable steps, and don't hesitate to review the examples and definitions as needed. Good luck!