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Rational Functions

Learn about rational functions as the quotient of polynomials, analyzing long-run behavior, leading terms, and asymptotes. Explore examples and applications to understand how different degrees impact the function’s characteristics.

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Rational Functions

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  1. Rational Functions Lesson 9.4

  2. Both polynomials Definition • Consider a function which is the quotient of two polynomials • Example:

  3. Long Run Behavior • Given • The long run (end) behavior is determined by the quotient of the leading terms • Leading term dominates forlarge values of x for polynomial • Leading terms dominate forthe quotient for extreme x

  4. Example • Given • Graph on calculator • Set window for -100 < x < 100, -5 < y < 5

  5. Example • Note the value for a large x • How does this relate to the leading terms?

  6. Try This One • Consider • Which terms dominate as x gets large • What happens to as x gets large? • Note: • Degree of denominator > degree numerator • Previous example they were equal

  7. When Numerator Has Larger Degree • Try • As x gets large, r(x) also gets large • But it is asymptotic to the line

  8. Summarize Given a rational function with leading terms • When m = n • Horizontal asymptote at • When m > n • Horizontal asymptote at 0 • When n – m = 1 • Diagonal asymptote

  9. Extra Information • When n – m = 2 • Function is asymptotic to a parabola • The parabola is • Why?

  10. Try It Out • Consider • What long range behavior do you predict? • What happens for large x (negative, positive) • What happens for numbers close to -4?

  11. Application • Cost to manufacture n units isC(n) = 5000 + 50n • Average cost per unit is • What is C(1)? C(1000)? • What is A(1)? A(1000)? • What is the trend for A(n) when n gets large?

  12. Assignment • Lesson 9.4 • Page 413 • Exercises 1 – 21 odd

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