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Chapter 2

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

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  1. Chapter 2 2-6 rational functions

  2. 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)

  3. objectives • Find the domains of rational functions • Find vertical and horizontal asymptotes of graphs • Use rational functions to model and solve real-life problems

  4. 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)

  5. 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

  6. What are asymptotes? • An asymptote is a line that the graph of a function approaches but never reaches.

  7. Types of asymptotes • There are two main types of asymptotes: Horizontal and Vertical .

  8. Vertical and horizontal asymptotes • What is vertical asymptote and horizontal asymptote?

  9. 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.

  10. 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.

  11. 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.

  12. 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

  13. 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

  14. Example#1 • The graph has a vertical asymptote at x=_____. • The Equation has horizontal asymptote of • Y=____

  15. 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

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

  17. 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

  18. Student guided practice • Do problems 1 -4 on the worksheet

  19. Homework • Do problems 17-20 and 25-28 from your book page 148

  20. 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

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