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Precalculus Lesson

Precalculus Lesson. Quiz 2.4-2.5 Today. Check: p.206 #1,3,17,23,25,31,35,41,45,49,59,61,63,67,71,77,79,91,95,103,107. Warm-up. Use the given zero to find all the zeros of the function. Answers. {2i, -2i, -1/2, 1}. Section 2.6. Date: _______

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Precalculus Lesson

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  1. PrecalculusLesson Quiz 2.4-2.5 Today Check: p.206 #1,3,17,23,25,31,35,41,45,49,59,61,63,67,71,77,79,91,95,103,107

  2. Warm-up Use the given zero to find all the zeros of the function.

  3. Answers {2i, -2i, -1/2, 1}

  4. Section 2.6 • Date: _______ • Objective: Analyze and sketch graphs of rational functions.

  5. Notes 2.6 Rational Functions and Asymptotes. • A Rational Function, where p(x) and q(x) have no common factors, can have several vertical asymptotes, but at most one horizontal asymptote. **Note—The graph may cross a horizontal asymptote, but not a vertical one!!

  6. To find a vertical asymptote; • Set the denominator equal to zero and solve for x.

  7. To find horizontal asymptotes: • Compare the degree of p(x) (the numerator) with the degree of q(x) (the denominator.)

  8. 1. Numerator degree < Denominator degree Horizontal asymptote; y=0 • Since 2<3, then y=0 is the horizontal asymptote.

  9. 2. Numerator degree = Denominator degreeHorizontal asymptote; **Note a and b represent the leading coefficients of the numerator and denominator. Since 2=2, then y=2/3 is the horizontal asymptote.

  10. 3. Numerator degree > Denominator degreeNo horizontal asymptote • Since 5>3, then there is no horizontal asymptote.

  11. Ex 1 Find the horizontal and vertical asymptotes. a) b) c)

  12. Slant Asymptotes • IF the numerator degree is exactly onemore than the denominator degree, then the function has a slant asymptote. Use synthetic or long division to find the slant asymptote.

  13. The quotient without the remainder is the slant asymptote. y=x-2 is the slant asymptote.

  14. Ex 2 Find all of the asymptotes and sketch the graph. A)

  15. Other items to find and label: • Domain –Must find first. Set Den=0 and solve to find values to exclude! • Zeros—Set the Numerator=0 and solve. Some “zeros” must be discarded because not in domain. • Hole in graph at discarded value. • Asymptotes--Reduce if necessary. • Y-coordinate for hole —evaluate reduced function at discarded value. F(discard)=??

  16. B) Find the Domain first, then reduce to find asymptotes. NOTE—There will be a hole in the graph for the factor that cancels. Label the asymptotes, zeros, and the hole in the graph.

  17. Assignments Classwork: p. 191 # 41, 47, 65 Homework(2.6): p. 190 #17-20 (matching), 26, 28, 42, 48, 62, 66, 90

  18. 333deer, 500 deer, 800 deer 1500 is the limiting size of the herd. [Horizontal Asymptote] The game commission introduces 100 deer into newly acquired state land. The population N of the herd is modeled by Where t is time in years. Find the populations when t = 5, t=10, t=25. What is the limiting size of the herd as time increases?

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