Name ______Date ______Period ______

Pre-Calculus

2.6: Rational Functions and Their Graphs

Rational Functions

  • ______are quotients of polynomial functions.
  • The ______of a rational expression is all real numbers except those that cause the ______to equal ______.

Example 1 (like HW #1-8)

Find the domain of

You-Try #1 (like HW #1-8)

Find the domain of

Arrow Notation

  • ______
  • “As approaches from the right, approaches infinity.”
  • ______
  • “As approaches from the left, approaches negative infinity.”
  • ______
  • “As approaches infinity, approaches zero.”

Vertical Asymptotes

  • An ______is a line that the graph of ______.
  • A graph can ______touch or cross a ______asymptote.
  • The line ______is a ______if increases or decreases without bound as ______approaches ______.
  • ______
  • ______
  • If _____ is a zero of ______but not a zero of ______, then ______is a vertical asymptote.

Example 2 (like HW #21-28)

Find the vertical asymptotes, if any, of

You-Try #2 (like HW #21-28)

Find the vertical asymptotes, if any, of

Holes

  • A ______is a point that is ______part of the ______of a function, but does ______cause an ______.
  • If ______is a zero of ______and a zero of ______, then there is a hole at ______.
  • Holes generally are ______distinguishable on a graphing calculator graph.

Example of a Hole

Horizontal Asymptotes

  • The line ______is a ______if approaches ______as increases or decreases without bound.
  • ______

OR

  • ______

Identifying Horizontal Asymptotes

  • Only the ______degree term of the top and bottom matter.
  • Let ______equal the degree of ______, the numerator.
  • Let ______equal the degree of ______, the denominator.
  • If ______, then the -axis () is a horizontal asymptote.
  • If ______, then the line is the horizontal asymptote.
  • If ______, then does ______have a horizontal asymptote.

Example 3 (like HW #29-33)

Find the horizontal asymptote, if any, of each function

You-Try #3 (like HW #29-33)

Find the horizontal asymptote, if any, of each function

Graphing Rational Functions

  1. Identify any ______asymptotes (numbers that are zeros of ______but not zeros of ______). Draw a dashed line for each.
  2. Identify any ______(x-values are numbers that are ______of both ______and ______.
  3. Identify any ______asymptotes by examining the ______terms. Draw a dashed line if one exists.
  4. Use the ______feature on your graphing calculator to get other ______to graph. Plot these on your graph.
  5. Draw a curve through the points, ______but not touching the ______. If there was a ______identified in step 2, put an ______at that -value.
  6. Check your graph with a graphing calculator. Remember that it does ______properly display asymptotes and holes.

Example 4 (like HW #37-58)

Graph

You-Try #4 (like HW #37-58)

Graph

You-Try #5 (like HW #37-58)

Graph

You-Try #6 (like HW #37-58)

Graph

Slant Asymptotes

  • A ______is a line of the form ______that the graph of a function approaches as ______
  • The graph of has a slant asymptote if the ______of the numerator is exactly ______than the ______of the denominator.
  • Find the equation of the slant asymptote by division (synthetic or long), and ______the remainder.

Example 7 (like HW #59-66)

Find the slant asymptote and graph

You-Try #7 (like HW #59-66)

Find the slant asymptote and graph

Applications of Rational Functions

  • The average cost of producing an item
  • Chemical concentrations over time
  • Used in numerous science and engineering fields to approximate or model complex situations.

Example 8 (page 332 #70)

The rational function describes the cost, , in millions of dollars, to inoculate of the population against a particular strain of the flu.

a)Find and interpret , , , , and .

b)What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function?

c)Graph the function.

2.6: Rational Functions and Their Graphs Page 1 of 9