Rational Functions and Their Graphs: Study Notes for College Algebra

Rational Functions and Their Graphs

Definition of Rational Functions

A rational function is any function that can be written as the ratio of two polynomials, that is:

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

• Rational functions are defined for all real numbers except where the denominator equals zero.

Domain of a Rational Function

The domain of a rational function consists of all real numbers except those for which the denominator is zero.

• To find the domain, solve $q(x) = 0$ and exclude those values from the domain.

• Example: For $f(x) = \frac{2x-25}{x-5}$, the domain is all real numbers except $x = 5$.

• Example: For $g(x) = \frac{25}{x^2-5}$, the domain is all real numbers except $x = \pm\sqrt{5}$.

• Example: For $h(x) = \frac{25}{x^2+5}$, the domain is all real numbers since $x^2+5$ is never zero.

Vertical Asymptotes

A vertical asymptote occurs at values of $x$ where the denominator of a rational function is zero (and the numerator is not zero at that point).

• To locate vertical asymptotes, solve $q(x) = 0$.

• Example: For $f(x) = \frac{x^2}{x-1}$, the vertical asymptote is at $x = 1$.

• Example: For $h(x) = \frac{x^2-1}{x+1}$, the vertical asymptote is at $x = -1$.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as $x$ approaches infinity or negative infinity. The line $y = b$ is a horizontal asymptote if $f(x)$ approaches $b$ as $x$ increases or decreases without bound.

• Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials.

Locating Horizontal Asymptotes

• If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$.

• If $n = m$, the horizontal asymptote is $y = \frac{a_n}{b_m}$, where $a_n$ and $b_m$ are the leading coefficients.

• If $n > m$, there is no horizontal asymptote.

Arrow Notation

Arrow notation is used to describe the behavior of functions as $x$ approaches certain values or infinity.

Symbol	Meaning
$x \to a^+$	$x$ approaches $a$ from the right.
$x \to a^-$	$x$ approaches $a$ from the left.
$x \to \infty$	$x$ approaches infinity; that is, $x$ increases without bound.
$x \to -\infty$	$x$ approaches negative infinity; that is, $x$ decreases without bound.

Strategy for Graphing Rational Functions

To graph a rational function, follow these steps:

1. Determine whether the graph has symmetry (y-axis or origin).

2. Find the y-intercept by evaluating $f(0)$.

3. Find the x-intercepts by solving $p(x) = 0$.

4. Find any vertical asymptotes by solving $q(x) = 0$.

5. Find the horizontal asymptote using the rules above.

6. Plot at least one point between and beyond each x-intercept and vertical asymptote.

7. Use the information to graph the function between and beyond the vertical asymptotes.

Slant (Oblique) Asymptotes

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. The slant asymptote can be found by performing polynomial long division.

• Example: For $f(x) = \frac{x^3-3x}{x^2-2}$, divide $x^3-3x$ by $x^2-2$ to find the slant asymptote.

Applications: Average Cost Function

Rational functions are used in business applications, such as average cost functions.

• Example: If a company has a fixed monthly cost of $500,000 and a variable cost of $400 per item, the total cost function is$C(x) = 500,000 + 400x$.

• The average cost function is $\frac{C(x)}{x} = \frac{500,000}{x} + 400$.

• As $x \to \infty$, the average cost approaches $400$, which is the horizontal asymptote.

• This represents the minimum average cost per item as production increases.

Graphing Practice

Use a blank Cartesian grid to practice graphing rational functions, marking intercepts and asymptotes.