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Understanding Perpendicular Lines and Rational Functions: SAT Question of the Day

In this lesson, we explore the relationship between perpendicular lines, particularly in relation to rational functions. Given that lines l and m intersect at the origin with line l passing through (2,-1), we analyze potential points line m could pass through, enhancing our understanding of geometrical positioning. Furthermore, we delve into the domains and asymptotes of rational functions, essential concepts for SAT preparation. Learn to identify vertical and horizontal asymptotes and apply these principles in real-life problems effectively.

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Understanding Perpendicular Lines and Rational Functions: SAT Question of the Day

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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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