Rational Functions and Their Graphs: Asymptotes and Applications

Rational Functions and Models

Definition of Rational Functions

A rational function is any function that can be written in the 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 those for which $q(x) = 0$ (since division by zero is undefined).

• Key Point: The domain excludes all values of $x$ that make the denominator zero.

• Example: $f(x) = \frac{2x+1}{x-3}$ has domain $x \neq 3$.

Determining Rational Functions and Their Domains

To determine if a function is rational, check if it is a ratio of two polynomials. For each rational function, set the denominator equal to zero and solve for $x$ to find excluded values from the domain.

• Example: $f(x) = \frac{x^2+1}{x^2-4}$ is rational. The domain is all real $x$ except $x = 2$ and $x = -2$.

Asymptotes of Rational Functions

Vertical Asymptotes

A vertical asymptote occurs at $x = k$ if $q(k) = 0$ and $p(k) \neq 0$. As $x$ approaches $k$ from the left or right, $f(x)$ increases or decreases without bound.

• Key Point: Vertical asymptotes are found by setting the denominator equal to zero and solving for $x$.

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x = 2$.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line $y = L$ that the graph of $f(x)$ approaches as $x$ goes to $\infty$ or $-\infty$. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator polynomials:

• If degree of numerator < degree of denominator: $y = 0$ is the horizontal asymptote.

• If degree of numerator = degree of denominator: $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.

• If degree of numerator > degree of denominator: There is no horizontal asymptote.

Identifying Asymptotes from Graphs

Vertical asymptotes appear as dashed vertical lines where the graph shoots up or down without bound. Horizontal asymptotes are dashed horizontal lines that the graph approaches as $x$ becomes very large or very small.

Step-by-Step: Finding Asymptotes

1. Factor numerator and denominator completely. Simplify if possible.

2. Vertical Asymptotes: Set denominator equal to zero and solve for $x$.

3. Horizontal Asymptotes: Compare degrees of numerator and denominator and apply the rules above.

Applications of Rational Functions

Population and Survival Models

Rational functions are often used to model real-world phenomena such as population growth or survival rates. For example, the population of rabbits on a farm or the percentage of trees surviving to a certain age can be modeled using rational functions.

• Example: $P(t) = \frac{40t + 20}{t + 5}$ models the population of rabbits $t$ months after introduction.

• To find $P(0)$ and $P(5)$:

  • $P(0) = \frac{40 \cdot 0 + 20}{0 + 5} = \frac{20}{5} = 4$

  • $P(5) = \frac{40 \cdot 5 + 20}{5 + 5} = \frac{200 + 20}{10} = \frac{220}{10} = 22$

• Interpretation: At $t=0$, there are 4 rabbits; at $t=5$ months, there are 22 rabbits.

Horizontal Asymptote in Context

The horizontal asymptote of a population model like $P(t) = \frac{40t + 20}{t + 5}$ can be found by comparing degrees:

• Both numerator and denominator are degree 1, so the horizontal asymptote is $y = \frac{40}{1} = 40$.

• Interpretation: As $t$ becomes very large, the population approaches 40 rabbits.

Vertical Asymptote in Context

For $P(t) = \frac{40t + 20}{t + 5}$, the vertical asymptote is at $t = -5$. In the context of time, negative values may not be meaningful, so the vertical asymptote may not have a practical interpretation for the population.

Graphical Examples

Summary Table: Asymptote Rules

Case	Horizontal Asymptote	Vertical Asymptote
deg(numerator) < deg(denominator)	$y = 0$	Set denominator = 0
deg(numerator) = deg(denominator)	$y = \frac{a}{b}$ (leading coefficients)	Set denominator = 0
deg(numerator) > deg(denominator)	None	Set denominator = 0

Additional info:

• When simplifying rational functions, always reduce to lowest terms before identifying asymptotes.

• Some rational functions may have holes (removable discontinuities) if a factor cancels from numerator and denominator.