Chapter 2

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# Chapter 2 - PowerPoint PPT Presentation

Chapter 2. 2-6 rational functions. SAT Question of the day . Lines l and m are perpendicular lines that intersect at the origin. If line l passes through the point (2,-1), then line m must pass through which of the following points? A)(0,2) B)(1,3) C)(2,1) D)(3,6) E)(4,0). objectives.

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### Chapter 2

2-6 rational functions

SAT Question of the day
• Lines l and m are perpendicular lines that intersect at the origin. If line l passes through the point (2,-1), then line m must pass through which of the following points?
• A)(0,2)
• B)(1,3)
• C)(2,1)
• D)(3,6)
• E)(4,0)
objectives
• Find the domains of rational functions
• Find vertical and horizontal asymptotes of graphs
• Use rational functions to model and solve real-life problems
What are rational functions?
• rational function is defined as the quotient of two polynomial functions.
• f(x) = P(x) / Q(x)
• Here are some examples of rational functions:
• g(x) = (x2 + 1) / (x - 1)
• h(x) = (2x + 1) / (x + 3)
Example#1
• Example: Find the domain of each function given below.
• g(x) = (x - 1) / (x - 2)
• h(x) = (x + 2) / x
• Solution
• For function g to be defined, the denominator x - 2 must be different from zero or x not equal to 2. Hence the domain of g is given by
• For function h to be defined, the denominator x must be different from zero or x not equal to 0. Hence the domain of h is given by
What are asymptotes?
• An asymptote is a line that the graph of a function approaches but never reaches.
Types of asymptotes
• There are two main types of asymptotes: Horizontal and Vertical .
Vertical and horizontal asymptotes
• What is vertical asymptote and horizontal asymptote?
Vertical asymptote
• Vertical Asymptotes of Rational Functions
• To find a vertical asymptote, set the denominator equal to 0 and solve for x.  If this value, a, is not a removable discontinuity, then x=a is a vertical asymptote.
Horizontal asymptotes
• 1.  To find a function's horizontal asymptotes, there are 3 situations.
• a.  The degree of the numerator is higher than the degree of the denominator.
• 1.  If this is the case, then there are no horizontal asymptotes.
• b.  The degree of the numerator is less than the degree of the denominator.
• 1.  If this is the case, then the horizontal asymptote is y=0.
Horizontal asymptote
• The degree of the numerator is the same as the degree of the denominator.
• 1.  If this is the case, then the horizontal asymptote is y = a/d where a is the coefficient in front of the highest degree in the numerator and d is the coefficient in front of the highest degree in the denominator.
Horizontal asymptotes
• The graph of f has at most one horizontal asymptote determine by comparing the degree of the of P(x) and Q(x)

n is the degree of the numerator

M is the degree of the denominator

• Id n< m then the graph has a line y=o as a horizontal asymptote
• If m=n then the graph has the line
• If n>m the graph has no horizontal asymptote
General rules
• In general, the procedure for asymptotes is the following:
• set the denominator equal to zero and solve
• the zeroes (if any) are the vertical asymptotes
• everything else is the domain
• compare the degrees of the numerator and the denominator
• if the degrees are the same, then you have a horizontal asymptote at y = (numerator's leading coefficient) / (denominator's leading coefficient)
• if the denominator's degree is greater (by any margin), then you have a horizontal asymptote at y = 0 (the x-axis)
• if the numerator's degree is greater (by a margin of 1), then you have a slant asymptote which you will find by doing long division
Example#1
• The graph has a vertical asymptote at x=_____.
• The Equation has horizontal asymptote of
• Y=____
Example#2
• Find the domain and all asymptotes of the following function:

domain:  vertical asymptotes:  x = ± 3/2horizontal asymptote:  y = 1/4

Example#3
• Find the domain and all asymptotes of the following function:
• domain:  all xvertical asymptotes:  nonehorizontal asymptote:  y = 0 (the x-axis)
Example#4

Special Case with a "Hole"

• Find the domain and all asymptotes of the following function:
• domain:  vertical asymptote: x=2
• Horizontal asymptote: None
Student guided practice
• Do problems 1 -4 on the worksheet
Homework
• Do problems 17-20 and 25-28 from your book page 148
closure
• Today we learned about finding domain and range.
• We also learned how to find the vertical and horizontal asymptotes.
• Next class we are going to learned about graphs of rational functions