Rational Functions and Their Graphs: Study Guide

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Rational Functions and Their Graphs

Definition of Rational Functions

A rational function is any function that can be written as the quotient of two polynomial functions. The general form is:

General Form: where and are polynomials and .
Example:

Domain of Rational Functions

The domain of a rational function consists of all real numbers except those that make the denominator zero.

To find the domain: Set and solve for . Exclude these values from the domain.
Example 1: Domain: All real numbers except .
Example 2: Domain: All real numbers except .

Behavior of Fractions Near Zero and Infinity

Understanding how rational functions behave as the denominator approaches certain values is crucial for graphing and analysis.

As the denominator becomes very large (positive or negative infinity), the fraction approaches zero.
As the denominator approaches zero, the fraction approaches positive or negative infinity.
Example: approaches 0 as or , and approaches or as from the right or left.

Arrow Notation

Arrow notation is used to describe how approaches certain values.

Symbol	Meaning
	approaches from the right
	approaches from the left
	increases without bound
	decreases without bound

Limits and Asymptotic Behavior

Limits describe the behavior of as approaches certain values.

As : approaches a value from the right of .
As : approaches a value from the left of .
As or : approaches its end behavior (often an asymptote).

Vertical Asymptotes and Holes

A vertical asymptote is a vertical line where the function approaches infinity. A hole occurs when a factor cancels in both numerator and denominator.

Vertical Asymptotes: Occur at values where and the factor does not cancel with the numerator.
Holes: Occur at values where a common factor cancels in both and .
Example: Vertical asymptote at .
Example: Factor numerator: . If denominator is , then is a hole if cancels.

Horizontal, Slant, and Non-Linear Asymptotes

Asymptotes describe the end behavior of rational functions.

Horizontal Asymptote: A horizontal line that approaches as or .
Slant (Oblique) Asymptote: Occurs when the degree of the numerator is one more than the denominator.
Non-Linear Asymptote: Occurs if the numerator's degree is more than one greater than the denominator.

Rules for Finding Horizontal Asymptotes:

If degree of numerator < degree of denominator , then is the horizontal asymptote.
If , then , where and are the leading coefficients.
If , there is no horizontal asymptote.

To find slant or non-linear asymptotes: Divide the denominator into the numerator and ignore the remainder.

Examples of Asymptotes

Example: Degree numerator (2) > degree denominator (1): Slant asymptote exists.
Example: Degree numerator (2) > degree denominator (1): Slant asymptote exists.
Example: Same as above.
Example: Degree numerator (3) > degree denominator (1): Non-linear asymptote.
Example: Degree numerator (4) > degree denominator (1): Non-linear asymptote.

Steps for Graphing a Rational Function

To graph a rational function , follow these steps:

Find the y-intercept: Evaluate .
Find the x-intercepts: Set and solve for .
Find vertical asymptotes: Set and solve for . Ensure no common factors with .
Find horizontal asymptote: Use degree comparison rules.
Find slant asymptote: If degree of numerator is one more than denominator, perform division.
Plot additional points: Find at least one point between and beyond each x-intercept and vertical asymptote.

Graphing Examples

Example: Find intercepts, asymptotes, and plot points.
Example: Find intercepts, asymptotes, and plot points.
Example: Find intercepts, asymptotes, and plot points.
Example: Find intercepts, asymptotes, and plot points.

Additional info: For each graph, follow the steps above to analyze intercepts, asymptotes, and behavior near critical points.