2.6Rational Functions

*Also referred to as the reciprocal function The standard form of a

rational function is:

f(x)= N(x) D(x)

*A function is rational if ‘x’ is in thedenominator, after the function has been simplified

What is a Rational Function ?

Are the following functions rational?

◦ Domain will generally be all real numbers except for the Vertical Asymptote(s) or holes of the function

◦ Range – generally all real numbers except for the Horizontal Asymptote or holes of the function

Finding Domain/Range of Rational Functions…

Domain continued…Ex:

-4 and -3 both make the denominator equal to zero, so they are both excluded from the domain

What is the domain of the following functions?1. f(x)=

2. f(x)=

Describing the domain…

Ex: f(x) =

Description:As x decreases to 0 y increases without bound,as x increases to 0 y decreases without bound.

Ex. 2: Describe the domain of f(x) =

Ex. 3: Describe the domain of f(x) =

Let’s practice…

Asymptote- the line that the function approaches, helps determine end behaviors.

*graphs will NEVER cross vertical asymptotes but they can cross horizontal or slant Holes - point of discontinuity (function is

undefined at this value)

What are Asymptotes?

Asymptotes

Finding Vertical Asymptotes and Holes…

Vertical Asymptote- set the denominator equal to zero.

Holes – occur when a factor in the denominator is simplified (reduced to 1) by same factor in the numerator

Ex 1:

Ex. 2:

Identify any vertical asymptotes or holes in the following rational functions:

Horizontal Asymptotes…

Case 1: N < D

◦ Horizontal asymptote: y = k *most scenarios, y = 0 Ex 1.

Ex 2.

Ex 3.

Cases of Horizontal Asymptotes

Case 2: N = D

◦ Horizontal asymptote is y = *the leading coefficients

◦ Ex. 1)

◦ Ex 2.)

Cases of Horizontal Asymptotes

Case 3: N > D

◦ Horizontal Asymptote is NONE *improper fraction!

◦ This case will have a SLANT Asymptote

◦ Ex. 1)

Cases of Horizontal Asymptotes

Pg 174 #’s 1, 5, 7, 11 (describe the domains as well for 5 - 11), 13-16, 17, 25, 39

Have fun

Homework Day 1:

Identify any vertical asymptotes, horizontal asymptotes, and/or holes for the following rational function

Describe the domain (using correct notation):

Day 2 – Warm Up: Use the following function:

An oblique line that the graph approaches, it helps describe/determine the end behaviors

If the degree of the numerator is exactly 1 more than the denominator it has a slant asymptote

What is a Slant Asymptote?

Check the Degree Use either long or synthetic division to find asymptote

Find the Slant Asymptote

◦ Ex 1.)

◦Ex 2.)

Finding Slant Asymptotes

f(x) =

f(x) =

f(x) =

Identify any asymptotes of the following rational functions…

Graphing Rational Functions

1. Find the zeros of the denominator. These will be the vertical asymptotes/holes. Draw dotted line(s).

2. Find the horizontal asymptotes in the ways learned earlier. Draw dotted line(s).

3. See if there will be any slant asymptotes in the graph. Draw dotted line(s).

4. Find the zeros of the numerator. These will be the x-intercepts unless it is also a zero of the denominator.

5. Evaluate f(0) to find y intercept of function. Plot.6. Create a table of values and plug in at least one point

between and one point beyond each x-intercept and vertical asymptote.

7. Connect points with smooth curves

Things to keep in mind:◦ Positive numerator, functions will be in quadrants 1 and 3◦ Negative numerator, functions will be in quadrants 2 and 4

Graph the following functions

Graphs of a Rational Function

1)

2)

3)

Some more practice with graphing…

Pg. 174-175 #’s 3(you don’t need

to fill in the tables), 19, 23, 35, 39, 45, 49, 51, 55, 69

Homework Day 2…