Polynomial and Rational Functions

1
Polynomial and Rational Functions

• Chapter 3

2
Properties of Rational Functions

• 3.3

3
Rational Function

• A function of the form
• R(x) p(x)/q(x)
• Where p and q are polynomial functions and q is
  not the zero polynomial.

4
Asymptotes

• Vertical or Horizontal lines that a graph
  approaches, but never reaches.

5
Finding Vertical Asymptotes of a Rational Function

• Find the x-values that make the denominator of
  the rational function in lowest terms zero.
  These x-values will be the zeros or factors of
  the polynomial in the denominator.

6
Homework

• Pg. 184 95 a,d,e,f,g
• Pg. 196-197 11-21 odd
• 41-51 odd VA only

7
Finding Horizontal Asymptotes of a Rational
Function

• Horizontal asymptotes occur as x approaches
  positive or negative infinity if they exist.

8
Finding Horizontal Asymptotes of a Rational
Function

• If the rational function p(x)/q(x), and the
  degree of p is the same as the degree of q, then
  a horizontal asymptote is found by making a
  fraction out of the coefficients of the highest
  degree terms of each polynomial.

9
Finding Horizontal Asymptotes of a Rational
Function

• If the rational function p(x)/q(x), and the
  degree of q is the larger than the degree of p,
  then the function will have a horizontal
  asymptote at zero.

10
Finding Horizontal Asymptotes of a Rational
Function

• If the rational function p(x)/q(x), and the
  degree of p is the larger than the degree of q,
  then the function will have no horizontal
  asymptote.

11
Using the Calculator to Find Horizontal Asymptotes

• Press 2nd Window to get to the table setup
  screen.
• Put in a large number for table start, such as
  1,000.
• Press 2nd Graph to go to the table and see the
  values displayed for y.
• Repeat with the table start at 10,000, etc.
• If the y values approach a real number b, then b
  is the horizontal asymptote.

12
Using the Calculator to Find Horizontal Asymptotes

• Press 2nd Window to get to the table setup
  screen.
• Put in a large number for table start, such as
• -1,000.
• Press 2nd Graph to go to the table and see the
  values displayed for y.
• Repeat with the table start at -10,000, etc.
• If the y values approach a real number b, then b
  is the horizontal asymptote.

13
Finding Oblique Asymptotes of a Rational Function

• If the rational function p(x)/q(x), and the
  degree of p is one larger than the degree of q,
  then the function will have an oblique asymptote.
  The quotient from long division will give the
  equation for the oblique asymptote.

14
Polynomial Long Division

• Set up the division so that dividend is written
  in descending power of x order, and has all
  powers of x represented.
• Divide the first term of the divisor by the first
  term of the dividend.
• Write the answer above the first term and then
  multiply it by the divisor.

15
Polynomial Long Division

• Write that answer under the positions that match
  the powers of x.
• Subtract
• Carry down all remaining terms.
• Divide the first term of this result by the first
  term of the divisor.
• Write this answer above the second term in the
  dividend.
• Multiply this answer by the divisor.

16
Polynomial Long Division

• Write that answer under the positions that match
  the powers of x.
• Subtract.
• Carry down all remaining terms.
• Repeat until the degree of this result is less
  than the degree of the divisor.

17
The Graph of a Rational Function Inverse and
Joint Variation

• 3.4

18
Homework

• Pg. 196-197 51
• Pg. 208 11,23,41
• In class pg. 208 7, 9 following the 8 steps
  outlined in the chapter on page 198.