Study Notes: Properties and Graphs of Rational Functions

Polynomial and Rational Functions

Definition of Rational Functions

A rational function is any function that can be written in the form $R(x) = \frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomial functions and q(x) is not the zero polynomial. The domain of a rational function is all real numbers except those for which the denominator is zero.

• Key Point: The domain excludes all real numbers that make $q(x) = 0$.

• Example: $R(x) = \frac{x^2 - 4}{x - 2}$ has domain $\{x \mid x \neq 2\}$.

Finding the Domain of a Rational Function

To find the domain, set the denominator equal to zero and solve for the excluded values.

• Example: $R(x) = \frac{1}{x^2 + 1}$ has domain all real numbers, since $x^2 + 1 = 0$ has no real solutions.

• Example: $R(x) = \frac{1}{x^2 - 25}$ has domain $\{x \mid x \neq 5, x \neq -5\}$.

Properties of Rational Functions

Vertical Asymptotes

A vertical asymptote occurs at $x = r$ if $r$ is a real zero of the denominator (after simplifying the function to lowest terms). The graph approaches infinity or negative infinity as $x$ approaches $r$.

• Theorem: If $x - r$ is a factor of $q(x)$ in lowest terms, then $x = r$ is a vertical asymptote.

• Example: $F(x) = \frac{x + 3}{x - 3}$ has a vertical asymptote at $x = 3$.

Multiplicity and Vertical Asymptotes

• If the multiplicity of the zero is odd, the graph approaches $+\infty$ on one side and $-\infty$ on the other.

• If the multiplicity is even, the graph approaches the same infinity on both sides.

Multiplicity	Graph Behavior
Odd	Approaches $+\infty$ on one side, $-\infty$ on the other
Even	Approaches same infinity on both sides

Horizontal and Oblique (Slant) Asymptotes

Asymptotes describe the end behavior of rational functions as $x \to \pm\infty$.

• Horizontal Asymptote: If $\lim_{x \to \pm\infty} R(x) = L$, then $y = L$ is a horizontal asymptote.

• Oblique (Slant) Asymptote: If $R(x)$ approaches a line $y = ax + b$ as $x \to \pm\infty$, then $y = ax + b$ is an oblique asymptote.

Degree of Numerator (n)	Degree of Denominator (m)	Asymptote
n < m	m	Horizontal, $y = 0$
n = m	m	Horizontal, $y = \frac{a_n}{b_m}$ (ratio of leading coefficients)
n = m + 1	m	Oblique, $y = ax + b$ (from polynomial division)
n \geq m + 2	m	No horizontal or oblique asymptote

Graphing Rational Functions

To analyze and graph a rational function, follow these steps:

1. Find the domain by determining where the denominator is zero.

2. Simplify the function to lowest terms.

3. Find intercepts: Set numerator to zero for x-intercepts; set $x = 0$ for y-intercept (if in domain).

4. Locate vertical asymptotes by finding real zeros of the denominator in lowest terms.

5. Determine horizontal or oblique asymptotes using degree comparison.

6. Check for holes (removable discontinuities) where factors cancel in numerator and denominator.

7. Sketch the graph using all the above information.

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

• Domain: $x \neq 0$

• x-intercepts: $x = -2, 2$

• y-intercept: None (since $x = 0$ is not in domain)

• Vertical asymptote: $x = 0$ (odd multiplicity)

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

Holes in the Graph

A hole (removable discontinuity) occurs at $x = a$ if both the numerator and denominator have a common factor $(x - a)$, and the function is undefined at $x = a$ but can be simplified elsewhere.

• Example: $R(x) = \frac{(x+3)(x-3)}{(x+3)(x-1)}$ has a hole at $x = -3$ and a vertical asymptote at $x = 1$.

Summary Table: Asymptote Types and Graph Behavior

Asymptote Type	Equation	Graph Behavior
Horizontal	$y = L$	As $x \to \pm\infty$, $R(x) \to L$
Vertical	$x = c$	As $x \to c$, $|R(x)| \to \infty$
Oblique	$y = ax + b$	As $x \to \pm\infty$, $R(x) \to ax + b$

Calculator Table Example

Calculator tables can help visualize the behavior of rational functions near asymptotes and holes.

Graphical Examples

• Even vs. Odd Multiplicity at Vertical Asymptotes:

• Transformation Example: Shifting and stretching rational functions can be visualized by comparing $H(x) = \frac{1}{x^2}$ and $H(x) = \frac{1}{(x-1)^2} - 2$.

• Multiple Asymptotes: Some rational functions have more than one vertical asymptote, depending on the denominator's factors.

Conclusion

Understanding rational functions involves analyzing their domains, intercepts, asymptotes, and possible holes. Mastery of these concepts is essential for graphing and interpreting rational functions in precalculus.