Rational Functions and Their Graphs: Study Notes

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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:

General form: , where and are polynomials and .
Example:

Domain of Rational Functions

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

To find the domain, set and solve for .
Example 1: Set . Domain: All real numbers except .
Example 2: Set or . Domain: All real numbers except and .

Behavior of Rational Functions Near Zero

For :
- As , .
- As , .
- As , .
- As , .

Arrow Notation

Arrow notation is used to describe the behavior of functions as approaches a certain value.

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

Vertical Asymptotes and Holes

A vertical asymptote is a vertical line where the function increases or decreases without bound as approaches .

Vertical asymptotes occur at values of that make the denominator zero, provided the numerator does not also become zero at that value.
If a factor cancels from both numerator and denominator, the graph has a hole at that -value instead of an asymptote.
Example: Factor numerator: The cancels, so there is a hole at .

Horizontal, Slant, and Non-Linear Asymptotes

Asymptotes describe the end behavior of rational functions.

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

To find equations for these asymptotes, divide the denominator into the numerator and ignore the remainder.

Rules for Finding Horizontal Asymptotes

Let = degree of numerator, = degree of denominator.
If , the horizontal asymptote is (the -axis).
If , the horizontal asymptote is , where and are the leading coefficients of the numerator and denominator, respectively.
If , there is no horizontal asymptote.

Example Table: Horizontal Asymptote Rules

Degree Relationship	Horizontal Asymptote


	None

Finding Slant Asymptotes

If the degree of the numerator is exactly one more than the denominator, perform polynomial long division. The quotient (without the remainder) is the equation of the slant asymptote.
Example: Divide into to get (ignore remainder). Slant asymptote: .

Steps for Graphing a Rational Function

To graph :

Find the y-intercept: Evaluate .
Find the x-intercepts: Set and solve for .
Find vertical asymptotes: Set and solve for (after canceling common factors).
Find horizontal asymptote: Use degree rules above.
Find slant asymptote: If applicable, use polynomial division.
Plot additional points: Choose -values between and beyond intercepts and asymptotes to determine the graph's shape.

Summary Table: Key Features of Rational Functions

Feature	How to Find
Domain	Set denominator ; exclude those -values
Vertical Asymptotes	Zeros of denominator (after canceling common factors)
Holes	Common factors in numerator and denominator
Horizontal Asymptote	Compare degrees of numerator and denominator
Slant Asymptote	Numerator degree is one more than denominator; use division
x-intercepts	Zeros of numerator (not canceled by denominator)
y-intercept	Evaluate

Examples and Applications

Example 1: Factor numerator: Simplifies to for ; hole at .
Example 2: Degree numerator = 2, denominator = 1; , so no horizontal asymptote. Perform division for slant asymptote.
Example 3: Degrees equal; horizontal asymptote at (leading coefficients both 1).

Additional info: For more complex rational functions, always check for simplification before determining asymptotes and holes. Graphing calculators or software can help visualize these features, but understanding the algebraic process is essential for exams.