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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.

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chapter 2

Chapter 2

2-6 rational functions

sat question of the day
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
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
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#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
What are asymptotes?
  • An asymptote is a line that the graph of a function approaches but never reaches.
types of asymptotes
Types of asymptotes
  • There are two main types of asymptotes: Horizontal and Vertical .
vertical and horizontal asymptotes
Vertical and horizontal asymptotes
  • What is vertical asymptote and horizontal asymptote?
vertical 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
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
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 asymptotes1
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
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 11
Example#1
  • The graph has a vertical asymptote at x=_____.
  • The Equation has horizontal asymptote of
  • Y=____
example 2
Example#2
  • Find the domain and all asymptotes of the following function:

Then the full answer is:

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

example 3
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
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
Student guided practice
  • Do problems 1 -4 on the worksheet
homework
Homework
  • Do problems 17-20 and 25-28 from your book page 148
closure
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