Rational Functions Notes

Intro to Graphing Rational Functions (h.k form)

• Objective: Learn how to graph rational functions in H, K form and identify vertical & horizontal asymptotes, domain, range, and intercepts.

Parent Function of a Rational Function

• The parent function for rational functions is $f(x) = \frac{1}{x}$.
• This represents inverse variation.

Transformations

• General form: $g(x) = \frac{a}{x-h} + k$
  • h-value: Moves the graph left or right.
  • k-value: Moves the graph up or down.
  • a-value: Affects reflection, compression, or stretching.

Table of Values for Parent Function

Restriction

• Rational functions have a restriction: division by zero is undefined.
• For $f(x) = \frac{1}{x}$, $x \neq 0$.
• Domain in interval notation: $(-\infty, 0) \cup (0, \infty)$

Asymptotes of Rational Functions

• Asymptotes are lines that the graph approaches closely but doesn't touch.

Vertical Asymptote (V.A.)

• A vertical line that the graph approaches.
• Located at $x = h$

Horizontal Asymptote (H.A.)

• A horizontal line that the graph approaches.
• Located at $y = k$

Graphing and Characteristics

• To graph, consider the transformations and asymptotes.

Characteristics to Determine

• Vertical Asymptote (V.A.)
• Horizontal Asymptote (H.A.)
• Domain
• Range

Finding the a-value

• Begin at the "new origin" (intersection of asymptotes).
• Move one unit to the right.
• Move up or down to a "nice" point on the graph.
• If you move up, 'a' is positive; if down, 'a' is negative.

General form of a rational function

• $f(x) = \frac{g(x)}{A(x)}$ , where g(x) and A(x) are polynomials and $A(x) \neq 0$
• Values can be considered a discontinuity to the graph.

Holes of a Function

• A hole occurs when a factor can be canceled out.
• Set the canceled factor equal to zero and solve for x.
• Substitute the x-value into the simplified expression to find the y-value.

Determining Horizontal Asymptotes Without a Graph

• Compare the degrees of the numerator and denominator.
  • If the degree is higher on the bottom, $y = 0$
  • If the degree is the same on the top and bottom, $y = \frac{a}{b}$, ratio of leading coefficients.
  • If the degree is higher on the top, there is no horizontal asymptote.

Slant Asymptotes

• Occur when the degree of the numerator is exactly one higher than the degree of the denominator.

Finding Intercepts Algebraically

• x-intercept: Set the simplified numerator equal to zero and solve for x (when y = 0).
• y-intercept: Substitute zero for x in the simplified function (when x = 0).
  *Note: Vertical asymptotes create restrictions on the domain.

Solving Rational Equations

Steps

1. Factor denominator if possible.
2. Check for restrictions on x.
3. Multiply each TERM by LCD.
4. Simplify.
5. SOLVE for x and check.