3'4 Notes

1
3.4 Notes

• Graphing Rational Functions

2
3.4 Notes

• Graphing rational functions
• Use limits to find the discontinuities and graph
  them.
• Use a t-table to determine the behavior between
  discontinuities.
• Plot the points in the t-table and sketch a
  smooth curve

3
Finding the discontinuities

• Find the coordinates of holes. Factor the
  function to see if the numerator and denominator
  have common factors. If so, apply the theorem.
• If is a common factor of the
  numerator and denominator of f(x), then
• is a hole.

4
Finding the discontinuities

• Find the equations of horizontal asymptotes. The
  function will have horizontal asymptotes if the
  degree of the numerator is less than or equal to
  the degree of the denominator. If this is true,
  apply the theorem.
• is a horizontal asymptote of f(x)
  if
• or if

5
Finding the discontinuities

• Find the equations of slant asymptotes. The
  function may have slant asymptotes if the degree
  of the numerator is one more than the degree of
  the denominator. If so, apply the theorem.
• The oblique line is a slant asymptote of f(x)
  if
• or if
• when f(x) is in quotient form.

6
Finding the discontinuities

• Find the equations of vertical asymptotes. The
  function may have vertical asymptotes if the
  denominator is zero for some values of x. If so,
  apply the theorem.
• is a vertical asymptote of f(x) if
• or if
• from the left or the right.

7
3.4 Notes

• Example Find the discontinuities of
• Check for holes
• is a hole.

8
3.4 Notes

• Example Find the discontinuities of
• Check for horizontal asymptotes
• The degree of the numerator is greater than the
  degree of the denominator. This rational
  function has no horizontal asymptotes.

9
3.4 Notes

• Example Find the discontinuities of
• Check for slant asymptotes
• The degree of the denominator is one greater
  than the degree of the numerator. This rational
  function may have a slant asymptote.

10
3.4 Notes

• Example Find the discontinuities of
• Check for slant asymptotes
• Divide to put into quotient form

11
3.4 Notes

• Check for slant asymptotes
• Take the limit as x approaches infinity
• may be a slant asymptote. (Its not, it is
  actually the graph of the function.)

12
3.4 Notes

• Example Find the discontinuities of
• Check for vertical asymptotes
• If x 2, the denominator would be zero.
• A rational function can have both vertical
  asymptotes and holes, but they will never have
  them at the same x value. This function has a
  hole at (2, 7), so it does not have any vertical
  asymptotes.

13
3.4 Notes

• Example Find the discontinuities of
• Hole at
• No horizontal, vertical asymptotes. Could have
  a slant asymptote at y x 5, but it doesnt.

14
3.4 Notes

• Example Find the discontinuities of
• Check for holes
• No common factors in the numerator and
  denominator. This function has no holes.

15
3.4 Notes

• Example Find the discontinuities of
• Check for horizontal asymptotes
• The degree of the numerator is less than the
  degree of the denominator. It could have a
  horizontal asymptote.

16
3.4 Notes

• Check for horizontal asymptotes
• y 0 is a horizontal asymptote.

17
3.4 Notes

• Example Find the discontinuities of
• Check for slant asymptotes
• The degree of the numerator is less than the
  degree of the denominator. It wont have a
  slant asymptote.

18
3.4 Notes

• Example Find the discontinuities of
• Check for vertical asymptotes
• If x 0 or x 4, the denominator would be
  zero.

19
3.4 Notes

• Check for vertical asymptotes
• x 0 is a vertical asymptote.
• x 4 is a vertical asymptote.

20
3.4 Notes

• Example Find the discontinuities of
• No holes. A horizontal asymptote at y 0, no
  slant asymptotes, and vertical asymptotes at
• x 0 and at x 4.

21
Homework

• Find the discontinuities of the following
  rational functions.
• 1.
• 2.