Rational Functions Lesson 9.4

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

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

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

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

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

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

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

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

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? x -100 -10 10 50 100 1000 G(x) x -4.2 -4.1 -4.01 -3.99 -3.9 -3.8 G(x)

Application Cost to manufacture n units is C(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?

Assignment Lesson 9.4 Page 413 Exercises 1 – 21 odd