Short Run Behavior of Rational Functions Lesson 9

Short Run Behavior of Rational Functions Lesson 9. 5

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

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

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

Drawing the Graph of a Rational Function • Check the long run behavior § Based on leading terms § Asymptotic to 0, to a/b, or to y=(a/b)x • Determine zeros of the numerator § These will be the zeros of the function • Determine the zeros of the denominator § This gives the vertical asymptotes • Consider

Given the Graph, Find the Function • Consider the graph given with tic marks = 1 • What are the zeros of the function? • What vertical asymptotes exist? • What horizontal asymptotes exist? • Now … what is the rational function?

Look for the Hole • What happens when both the numerator and denominator are 0 at the same place? • Consider • We end up with indeterminate which is § Thus the function has a point for which it is not defined … a “hole”

Look for the Hole • Note that when graphed and traced at x = -2, the calculator shows no value • Note also, that it does not display a gap in the line

Assignment • Lesson 9. 5 • Page 420 • Exercises 1 – 41 EOO