Rational Functions and Models Lesson 4.5

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

Vertical Asymptotes A vertical asymptote happens when the function R(x) is not defined This happens when the denominator is zero Thus we look for the roots of the denominator Where does this happen for r(x)?

Vertical Asymptotes Finding the roots of the denominator View the graph to verify

Zeros of Rational Functions We know that So we look for the zeros of P(x), the numerator Consider What are the roots of the numerator? Graph the function to double check

Zeros of Rational Functions Note the zeros of the function when graphed r(x) = 0 when x = ± 3

Summary The zeros of r(x) are where the numerator has zeros The vertical asymptotes of r(x) are where the denominator has zeros

Assignment Lesson 4.5A Page 297 Exercises 1 – 61 Odd

This is a power function Direct Variation The variable y is directly proportional to x when: y = k * x (k is some constant value) Alternatively As x gets larger, y must also get larger keeps the resulting k the same This is a power function

Direct Variation Example: The harder you hit the baseball The farther it travels Distance hit is directly proportional to the force of the hit

Direct Variation Suppose the constant of proportionality is 4 Then y = 4 * x What does the graph of this function look like?

Inverse Variation The variable y is inversely proportional to x when Alternatively y = k * x -1 As x gets larger, y must get smaller to keep the resulting k the same

Inverse Variation Example: If you bake cookies at a higher temperature, they take less time Time is inversely proportional to temperature

Inverse Variation Consider what the graph looks like Let the constant or proportionality k = 4 Then

Assignment Lesson 4.5B Page 299 Exercises 63 – 91 odd