Polynomial and Rational Functions

Introduction to Rational Functions

Rational functions are a central topic in precalculus, involving ratios of polynomials. Their graphs often display unique features such as asymptotes and discontinuities, which are essential for understanding their behavior and applications.

• Definition: A rational function is any function of the form , where and are polynomials and .

• Domain: All real numbers except those for which .

• Discontinuity: Rational functions are often discontinuous at points where the denominator is zero.

The Reciprocal Function

Definition and Basic Properties

The reciprocal function is the simplest rational function with a variable denominator, given by .

• Domain:

• Range:

• Discontinuity: The function is undefined at .

• Asymptotes: The y-axis () is a vertical asymptote, and the x-axis () is a horizontal asymptote.

Graph and Symmetry

• Odd Function: , so the graph is symmetric with respect to the origin.

Transformations of the Reciprocal Function

Vertical and Horizontal Shifts

• Multiplying by a constant stretches or compresses the graph.

• Negative signs reflect the graph across the axes.

• Shifting the input, as in , moves the graph horizontally.

The Function

Definition and Properties

This rational function is defined for all real numbers except and is always positive.

• Domain:

• Range:

• Even Function: , so the graph is symmetric with respect to the y-axis.

• Asymptotes: The y-axis () is a vertical asymptote, and the x-axis () is a horizontal asymptote.

Graphing Rational Functions

General Steps for Graphing

To sketch the graph of a rational function , follow these steps:

1. Find any vertical asymptotes by setting and solving for .

2. Find any horizontal or oblique asymptotes by comparing the degrees of and .

3. Find the y-intercept by evaluating .

4. Find the x-intercepts by solving .

5. Determine if the graph intersects its nonvertical asymptote by solving or .

6. Plot additional points as needed to understand the graph's behavior in each interval.

7. Complete the sketch, noting intervals of increase and decrease.

Example: Graphing

• Vertical asymptotes at and .

• Horizontal asymptote at (since degree of numerator < denominator).

• Find intercepts and plot points in each interval.

Asymptotes of Rational Functions

Types of Asymptotes

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

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator:

  • If degree numerator < degree denominator:

  • If degrees are equal: , where and are leading coefficients

  • If degree numerator is one more than denominator: Oblique (slant) asymptote, found by polynomial division

Example: Finding Asymptotes

• For :

  • Vertical asymptote:

  • Oblique asymptote: (from division)

Points of Discontinuity (Holes)

If a rational function is not in lowest terms, a factor that cancels in both numerator and denominator creates a hole in the graph at that x-value.

• Example: has a hole at .

Applications: Rational Models

Example: Modeling Traffic Intensity

Rational functions can model real-world phenomena, such as traffic flow. For example, the average number of vehicles waiting in line at a parking ramp can be modeled by a rational function of the form , where is the traffic intensity (arrival rate/admittance rate).

• As approaches 1, increases rapidly, indicating long wait times as the system nears capacity.

Summary Table: Types of Asymptotes

Key Takeaways

• Rational functions are quotients of polynomials with important features such as asymptotes and discontinuities.

• Graphing involves identifying asymptotes, intercepts, and behavior near undefined points.

• Transformations and shifts affect the position and orientation of the graph.

• Applications include modeling real-world systems with constraints and rates.