3.4 Analyzing and Graphing Rational Functions in Precalculus

Rational Functions

Definition and Structure

Rational functions are a central topic in precalculus, defined as functions that can be expressed as the ratio of two polynomials. The general form is:

• Definition: A rational function is any function of the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Domain: The domain of a rational function is all real numbers except where $Q(x) = 0$.

• Examples: $R(x) = \frac{2x}{x^2-4}$, $f(x) = \frac{x^2-1}{x}$

Steps to Analyze and Graph a Rational Function

To graph a rational function, it is helpful to follow a systematic approach:

1. Write in Lowest Terms: Simplify the function by factoring and reducing numerator and denominator.

2. Find the Domain: Set the denominator equal to zero and solve for excluded values.

3. Locate x- and y-intercepts:

   • x-intercepts: Set the numerator equal to zero and solve for $x$.

   • y-intercept: Evaluate $R(0)$ if defined.

4. Locate Vertical Asymptotes: Set the denominator equal to zero and solve for $x$ values where the function is undefined.

5. Locate Horizontal or Oblique Asymptotes:

   • Horizontal Asymptote: Compare degrees of numerator and denominator.

   • Oblique (Slant) Asymptote: If degree of numerator is one more than denominator, use polynomial division.

6. Explore End Behavior: Analyze the function as $x \to \infty$ and $x \to -\infty$.

7. Sketch the Graph: Plot intercepts, asymptotes, and key points. Draw the curve respecting asymptotic behavior.

Key Properties of Rational Functions

• Vertical Asymptotes: Occur at zeros of the denominator (after simplification).

• Horizontal Asymptotes:

  • If degree of numerator < degree of denominator: $y=0$ is the horizontal asymptote.

  • If degrees are equal: $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

  • If degree of numerator > degree of denominator: No horizontal asymptote; may have an oblique asymptote.

• Removable Discontinuities: If a factor cancels in numerator and denominator, there is a hole in the graph at that $x$ value.

Examples

• Example 1: $R(x) = \frac{2x}{x^2-4}$

  • Domain: $x \neq 2, x \neq -2$

  • x-intercept: $x=0$

  • y-intercept: $R(0)=0$

  • Vertical asymptotes: $x=2$, $x=-2$

  • Horizontal asymptote: $y=0$ (degree numerator < denominator)

• Example 2: $f(x) = \frac{x^2-1}{x}$

  • Domain: $x \neq 0$

  • x-intercepts: $x=1$, $x=-1$

  • y-intercept: $f(0)$ undefined

  • Vertical asymptote: $x=0$

  • Oblique asymptote: $y=x$ (since degree numerator is one more than denominator)

Graphing Practice

• Sketch the graphs of the following functions using the steps above:

  • $y = \frac{3x^2-3x}{x^2+x-12}$

  • $g(x) = \frac{x^2-16x+2}{x^2-4}$

• Use the provided grid to plot intercepts, asymptotes, and sample points.

Partner Activity: Creating Rational Functions

Designing Functions with Specific Characteristics

Students are asked to create a rational function with the following properties:

• Crosses the x-axis at $x=2$

• Bounces on the x-axis at $x=1$

• Vertical asymptotes at $x=5$ and $x=6$

• Horizontal asymptote at $y=3$

To achieve these characteristics:

• Include $(x-2)$ in the numerator for a zero at $x=2$

• Include $(x-1)^2$ in the numerator for a bounce at $x=1$

• Include $(x-5)(x-6)$ in the denominator for vertical asymptotes

• Set the leading coefficient ratio to 3 for the horizontal asymptote

Example Construction:

• $f(x) = \frac{3(x-2)(x-1)^2}{(x-5)(x-6)}$

Compare and contrast with other group functions to analyze similarities and differences in graph behavior.

Optimization Applications (Additional info)

Least Cost and Minimizing Surface Area

Although primarily a calculus topic, some problems involve expressing cost or surface area as a function, which can be relevant for precalculus students learning to model with functions.

• Express cost or area in terms of a variable (e.g., radius or side length).

• Use algebraic manipulation to simplify and analyze the function.

Example: Finding the least cost of a can or minimizing the surface area of a box by expressing the quantity as a function and analyzing its behavior.

Additional info: Optimization problems are typically explored further in calculus, but the algebraic setup is a valuable precalculus skill.

Summary Table: Steps for Graphing Rational Functions

Step	Description
1. Simplify	Write the function in lowest terms
2. Domain	Find where the denominator is zero
3. Intercepts	Find x- and y-intercepts
4. Vertical Asymptotes	Set denominator equal to zero
5. Horizontal/Oblique Asymptotes	Compare degrees of numerator and denominator
6. End Behavior	Analyze as $x \to \infty$ and $x \to -\infty$
7. Sketch	Plot intercepts, asymptotes, and sample points