Warm UP! Factor the following:

Rational Functions LG 7-1: Characteristics & Graphs of Rational (quiz 12/5) LG 7-2: Inverses of Rational Functions (quiz 12/12)

Rational Functions General Equation: S and T are polynomial functions Verbally: f(x) is a rational function of x. Features: A rational function has a discontinuities - asymptotes and/or holes.

The parent rational function is: The shape is made by the behavior of a function as it approaches asymptotes. ALL rational functions will look similar to this parent graph as they are all part of the same family. An example of a rational function is:

How to find the Characteristics of Rational Functions Domain & Range Intercepts Discontinuities - Vertical Asymptotes, Horizontal Asymptotes, Slant/Oblique Asymptotes, and Holes

All rational functions have asymptotes in their graphs.

X-intercepts Where the function crosses the x-axis. A function can have none, one, or multiples To find the x-int of a Rational Function, set the numerator equal to zero and solve for x.

Find all x-intercepts of each function. Practice Find all x-intercepts of each function.

y-intercepts Where the function crosses the y-axis A function can have NO MORE THAN 1! To find the y-int of Rational Functions, substitute 0 for x.

Find all y-intercepts of each function. Practice Find all y-intercepts of each function.

Vertical Asymptotes A vertical asymptote is an invisible line that the graph will NEVER cross. The function is undefined at a VA. You can find the VA by setting the denominator equal to zero and solving for x.

Domain of a Rational Function The domain of a rational function is all real numbers excluding the discontinuities (the vertical asymptote) and the x-value of the hole (more tomorrow!)

The functions p and q are polynomials. Defn: Rational Function   The functions p and q are polynomials. The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.      

Domain of a Rational Function         {x | x  –4} or (-, -4)  (-4, )

Domain of a Rational Function         {x | x  2} or (-, 2)  (2, )

Domain of a Rational Function           {x | x  –3, 3} or (-, -3)  (-3, 3)  (3, )

When you have one vertical asymptote… Your graph is separated into 2 sections… To the left of the asymptote To the right of the asymptote

When you have two vertical asymptotes… Your graph is separated into 3 sections… To the left of both asymptotes In between the asymptotes To the right of both asymptotes

Find the Vertical Asymptotes & Domain: Practice Find the Vertical Asymptotes & Domain:

Horizontal Asymptotes of Rational Functions 3.6: Rational Functions and Their Graphs Horizontal Asymptotes of Rational Functions If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem. Locating Horizontal Asymptotes Let f be the rational function given by The degree of the numerator is n. The degree of the denominator is m. If n < m, the x-axis is the horizontal asymptote of the graph of f. If n = m, the line y = is the horizontal asymptote of the graph of f. If n > m, the graph of f has no horizontal asymptote.

A horizontal asymptote exists at y = 5/2. Asymptotes Horizontal Asymptote Example     A horizontal asymptote exists at y = 5/2.   A horizontal asymptote exists at y = 0.  

Find the Horizontal Asymptotes & Range: Practice Find the Horizontal Asymptotes & Range:

How do you find asymptotes in rational functions? Vertical Asymptotes Horizontal Asymptotes Compare the degrees of the numerator and the denominator 1. Set denominator equal to zero. 1. If bigger on BOTTOM then there is a HA at y = 0. 2. Solve for x 2. If n = m, then there is a HA at y = ___ (find by divide the leading coefficients). 3. If bigger on TOP, then there is no horizontal asymptotes (this means there could be a SLANT/OBLIQUE asymptote which we will discuss tomorrow!

Find all asymptotes of Vertical: x = -1 and x = 2 Horizontal: Degree of top = 1 Degree of bottom = 2 BIGGER on BOTTOM y = 0

Degree of denominator = 1 Find all asymptotes of Vertical: x = 0 Horizontal: Degree of numerator = 1 Degree of denominator = 1 EQUAL

No horizontal asymptote Find all asymptotes of Vertical: x - 1 = 0 x = 1 Degree of numerator = 2 Horizontal: Degree of denominator = 1 Bigger on TOP No horizontal asymptote

Find the following characteristics for the rational function below: For example: Find the following characteristics for the rational function below: Domain: Range: x-intercepts: y-intercepts: Horizontal asymptote: Vertical asymptotes:

Ex. 4 Find all asymptotes of and graph Vertical: x + 1 = 0 x = -1 Set denominator equal to zero x = -1 Horizontal: n > m by exactly one n = 2 m = 1 No horizontal asymptote Compare degrees Slant: Use long division

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