2.6 Rational Functions JMerrill,2010

Domain Find the domain of Think: what numbers can I put in for x???? Denominator can’t equal 0 (it is undefined there)

You Do: Domain Find the domain of Denominator can’t equal 0

You Do: Domain Find the domain of Denominator can’t equal 0

Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below. none

Vertical Asymptotes The figure below shows the graph of The equation of the vertical asymptote is

Vertical Asymptotes Definition: The line x = a is a vertical asymptote of the graph of f(x) if or as x a from either the left or the right. Look at the table of values for

Vertical Asymptotes As x approaches____ from the _______, f(x) -3 -1 -2.5 -2 -2.1 -10 -2.01 -100 -2.001 -1000 x f(x) -1 1 -1.5 2 -1.9 10 -1.99 100 -1.999 1000 As x approaches____ from the _______, f(x) approaches _______. As x approaches____ from the _______, f(x) approaches _______. -2 -2 right left Therefore, by definition, there is a vertical asymptote at

Therefore, a vertical asymptote occurs at x = -3. Vertical Asymptotes Describe what is happening to x and determine if a vertical asymptote exists, given the following information: x f(x) -2 1 -2.5 2.2222 -2.9 11.837 -2.99 119.84 -2.999 1199.8 x f(x) -4 -1.333 -3.5 -2.545 -3.1 -12.16 -3.01 -120.2 -3.001 -1200 Therefore, a vertical asymptote occurs at x = -3. As x approaches____ from the _______, f(x) approaches _______. As x approaches____ from the _______, f(x) approaches _______. -3 -3 left right

Vertical Asymptotes Set denominator = 0; solve for x Substitute x-values into numerator. The values for which the numerator ≠ 0 are the vertical asymptotes

Example What is the domain? x ≠ 2 so What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into numerator, if it ≠ 0, then it’s a vertical asymptote)

You Do Domain: x2 + x – 2 = 0 Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0, so x ≠ -2, 1 Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1

The graph of a rational function NEVER crosses a vertical asymptote The graph of a rational function NEVER crosses a vertical asymptote. Why? Look at the last example: Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!

Horizontal Asymptotes Definition:The line y = b is a horizontal asymptote if as or Look at the table of values for

Horizontal Asymptotes x f(x) -1 1 -10 -0.125 -100 -0.0102 -1000 -0.001 x f(x) 1 .3333 10 .08333 100 .0098 1000 .0009 y→_____ as x→________ y→____ as x→________ Therefore, by definition, there is a horizontal asymptote at y = 0.

Examples Horizontal Asymptote at y = 0 Horizontal Asymptote at y = 0 What relationships exists between the numerator and the denominator in each of these problems? The degree of the denominator is larger than the degree of the numerator.

Examples Horizontal Asymptote at y = 2 Horizontal Asymptote at What relationships exists between the numerator and the denominator in each of these problems? The degree of the numerator is the same as the degree or the denominator.

Examples No Horizontal Asymptote No Horizontal Asymptote What relationships exists between the numerator and the denominator in each of these problems? The degree of the numerator is larger than the degree of the denominator.

Asymptotes: Summary 1. The graph of f has vertical asymptotes at the _________ of the denominator.  2. The graph of f has at most one horizontal asymptote, as follows:  a)   If n < d, then the ____________ is a horizontal asymptote. b)    If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.) c)   If n > d, then the graph of f has ______ horizontal asymptote. zeros line y = 0 no

Asymptotes Some things to note: Horizontal asymptotes describe the behavior at the ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends. The graph of a rational function may or may not cross a horizontal asymptote. The graph of a rational function NEVER crosses a vertical asymptote. Why?

You Do Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: x = -1 Horizontal Asymptote: y = 2

You Do - Again Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: none Horizontal Asymptote: y = 0

The Last You-Do (for now :) Find all asymptotes of the following function Vertical Asymptote: x = 2 Horizontal Asymptote: none

Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes. The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both. When doing long division, we do not care about the remainder.

Example Find all asymptotes. Vertical Horizontal Slant none x = 1 y = x

Example Find all asymptotes: Vertical asymptote at x = 1 y = x + 1 n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.

Graphing Rational Functions To graph a rational function, you find all asymptotes You must show your work You must identify the domain and range You must identify the x- and/or y-intercepts You may have to “blow up” part of the graph (Zoom:Box) to actually see how the graph fits next to the asymptote.