Rational Functions and Asymptotes: Precalculus Study Notes (Section 3.3)

Rational Functions

Definition and Structure

A rational function is a function that can be expressed as the ratio of two polynomials:

• General form: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• The domain of a rational function is all real numbers except where the denominator $Q(x)$ is zero.

Example: $f(x) = \frac{1}{x}$

Additional info: Rational functions are undefined at values of $x$ that make the denominator zero.

Parent Rational Function and Transformations

The most basic rational function is $f(x) = \frac{1}{x}$, known as the "parent" rational function.

• Graph features: Two branches, one in each quadrant, approaching the axes but never touching them.

• Asymptotes: The lines $x = 0$ (vertical) and $y = 0$ (horizontal) are asymptotes.

• Transformations: Shifting, stretching, or reflecting the graph can be achieved by modifying the function, e.g., $g(x) = \frac{1}{x} + 1$ shifts the graph up by 1 unit.

Example: $g(x) = \frac{1}{x} + 1$

Application: Transformations help in sketching and understanding the behavior of rational functions.

Asymptotes of Rational Functions

Vertical Asymptotes

Vertical asymptotes occur at values of $x$ that make the denominator zero (unless the numerator is also zero at that point, which may create a hole).

• To find: Set $Q(x) = 0$ and solve for $x$.

• Example: For $R(x) = \frac{1}{x}$, the vertical asymptote is at $x = 0$.

• Example: For $G(x) = \frac{1}{x^2 - 4}$, set $x^2 - 4 = 0$ to get $x = 2$ and $x = -2$.

Horizontal and Oblique (Slant) Asymptotes

Horizontal and oblique asymptotes describe the end behavior of rational functions as $x$ approaches infinity or negative infinity.

• Horizontal Asymptote: Occurs when the degree of the numerator is less than or equal to the degree of the denominator.

• Oblique (Slant) Asymptote: Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Case	Asymptote Type	Equation
Degree numerator < degree denominator	Horizontal	$y = 0$
Degree numerator = degree denominator	Horizontal	$y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
Degree numerator = degree denominator + 1	Oblique (Slant)	Use long division to find the equation

Example: For $f(x) = \frac{2x^2 + 3}{x^2 - 1}$, degrees are equal, so horizontal asymptote is $y = \frac{2}{1} = 2$.

Example: For $h(x) = \frac{3x^2 - x}{x^2 + x + 1}$, degrees are equal, so horizontal asymptote is $y = 3$.

Example: For $g(x) = \frac{2x^2 + x + 2}{x - 1}$, numerator degree is one more than denominator, so use long division for oblique asymptote.

Finding Oblique Asymptotes

Oblique asymptotes are found by dividing the numerator by the denominator using polynomial long division.

• Example 1: $h(x) = \frac{3x^2 - x}{x^2 + x + 1}$ (degrees equal, horizontal asymptote at $y = 3$)

• Example 2: $g(x) = \frac{2x^2 + x + 2}{x - 1}$ (degree numerator is one more, use long division)

Additional info: The quotient (ignoring the remainder) from long division gives the equation of the oblique asymptote.

Summary Table: Types of Asymptotes

Type	How to Find	Example
Vertical	Set denominator equal to zero	$f(x) = \frac{1}{x}$, $x = 0$
Horizontal	Compare degrees of numerator and denominator	$f(x) = \frac{2x^2 + 3}{x^2 - 1}$, $y = 2$
Oblique (Slant)	Long division (degree numerator = degree denominator + 1)	$g(x) = \frac{2x^2 + x + 2}{x - 1}$

Practice and Homework

• Practice finding vertical, horizontal, and oblique asymptotes for various rational functions.

• Homework: Textbook pages 196, 197, problems 8-17 (odds), 30-36 (evens), 43-53 (odds).