1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

2. 3.5 Rational Functions: Graphs, Applications, and Models
The Reciprocal Function
The Function
Asymptotes
Steps for Graphing Rational Functions
Rational Function Models

3. Rational function
A function  of the form
where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

4. Rational Function Some examples of rational functions are
Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.

5. The Reciprocal Function
The simplest rational function with a variable denominator is the reciprocal function, defined by

6. The Reciprocal Function
The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of for some values of (x) for some values of x very close to 0. We use the table feature of a graphing calculator to do this.
The tables suggest that (x) gets larger and larger as x gets closer and closer to 0, which is written in symbols as

7. The Reciprocal Function
(The symbol x  0 means that x approaches 0, without necessarily ever being equal to 0.) Since x cannot equal 0, the graph of will never intersect the vertical line x = 0.This line is called a vertical asymptote.
On the other hand, as x gets larger and larger, the values of get closer and closer to 0, as shown in the tables. Letting x get larger and larger without bound (written x  ) causes the graph to move closer and closer to the horizontal line. This line is called a horizontal asymptote.

8. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (– , 0)  (0, )x y
– 2
– ½
– 1
undefined 2 1
decreases on the intervals
(–,0) and (0, ).

9. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (– , 0)  (0, )x y
– 2
– ½
– 1
undefined 2 1
It is discontinuous at x = 0.

10. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (– , 0)  (0, )x y
– 2
– ½
– 1
undefined 2 1
The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

11. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (– , 0)  (0, )x y
– 2
– ½
– 1
undefined 2 1
It is an odd function, and its graph is symmetric with respect to the origin.

12. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 1
Graph Give the domain and range.
Solution
The expression can be written as or indicating that the graph may be obtained by stretching the graph of vertically by a factor of 2 and reflecting it across either the y-axis or x-axis. The x- and y-axes remain the horizontal and vertical asymptotes. The domain and range are both still (– , 0)  (0, ).

13. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 2
Graph Give the domain and range.
Solution
The expression can be written as indicating that the graph may be obtained by shifting the graph of
to the left 1 unit and stretching
it vertically by a factor of 2.

14. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 2
Graph Give the domain and range.
Solution
The horizontal shift affects the domain, which is now
(– , – 1)  (– 1, ) .
The line x = – 1 is the vertical asymptote, and the line y = 0
(the x-axis) remains the horizontal asymptote. The range is still (– , 0)  (0, ).

15. The Function The Function The rational function defined by
also has domain (– , 0)  (0, ). We can use the table feature of a graphing
calculator to examine values of (x) for some x-values close to 0.

16. The Function
The tables suggest that (x) gets larger and larger as x gets closer and closer to 0. Notice that as x approaches 0 from either side, function values are all positive and there is symmetry with respect to the y-axis. Thus, (x)   as x  0. The y-axis
(x = 0) is the vertical asymptote.

17. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (0, ) x y  3
 2
 1 1
 ½ 4
 ¼ 16
undefined
increases on the interval (–,0) and decreases on the interval (0, ).

18. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (0, ) x y  3
 2
 1 1
 ½ 4
 ¼ 16
undefined
It is discontinuous at x = 0.

19. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (0, ) x y  3
 2
 1 1
 ½ 4
 ¼ 16
undefined
The y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

20. RECIPROCAL FUNCTION Domain: (– , 0)  (0, ) Range: (0, ) x y  3
 2
 1 1
 ½ 4
 ¼ 16
undefined
It is an even function, and its graph is symmetric with respect to the y-axis.

21. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 3
Graph Give the domain and range.
Solution The equation is equivalent to
where This indicates that the graph will be shifted 2 units to the left and 1 unit down.

22. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 3
Graph Give the domain and range.
Solution The equation is equivalent to
The horizontal shift affects the domain, which is now (– , – 2)  (– 2, ), while the vertical shift affects the range, now (– 1, ).

23. Graph Give the domain and range.
GRAPHING A RATIONAL FUNCTION
Example 3
Graph Give the domain and range.
Solution The equation is equivalent to
The vertical asymptote has equation x = – 2, and the horizontal asymptote has equation y = – 1.

24. Asymptotes
Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b:
1. If (x)   as x  a, then the line x = a is a vertical asymptote.
2. If (x)  b as x  , then the line y = b is a horizontal asymptote.

25. Determining Asymptotes
To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures.
1. Vertical Asymptotes
Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

26. Determining Asymptotes
2. Other Asymptotes
Determine any other asymptotes. Consider three possibilities:
(a) If the numerator has lower degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).

27. Determining Asymptotes
2. Other Asymptotes
Determine any other asymptotes. Consider three possibilities:
(b) If the numerator and denominator have the same degree, and the function is of the form
where an, bn ≠ 0,
then the horizontal asymptote has equation

28. Determining Asymptotes
2. Other Asymptotes
Determine any other asymptotes. Consider three possibilities:
(c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

29. Motion Problems
Note The graph of a rational function may have more than one vertical asymptote, or it may have none at all. The graph cannot intersect any vertical
asymptote. There can be at most one other (nonvertical) asymptote, and the graph may intersect that asymptote as we shall see in Example 7.

30. For each rational function , find all asymptotes.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
For each rational function , find all asymptotes. a.
Solution To find the vertical asymptotes, set the denominator equal to 0 and solve.
Zero-property
Solve each equation.

31. The equations of the vertical asymptotes are x = ½ and x = – 3.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
The equations of the vertical asymptotes are x = ½ and x = – 3.
To find the equation of the horizontal asymptote, divide each term by the greatest power of x in the expression. First, multiply the factors in the denominator.

32. Stop here. Leave the expression in complex form.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
Now divide each term in the numerator and denominator by x2 since 2 is the greatest power of x.
Stop here. Leave the expression in complex form.

33. As x gets larger and larger, the quotients
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
As x gets larger and larger, the quotients
all approach 0, and the value of (x) approaches
The line y = 0 (that is, the x-axis) is therefore the horizontal asymptote.

34. For each rational function , find all asymptotes.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
For each rational function , find all asymptotes. b.
Solution Set the denominator x – 3 = 0 equal to 0 to find that the vertical asymptote has equation x = 3. To find the horizontal asymptote, divide each term in the rational expression by x since the greatest power of x in the expression is 1.

35. For each rational function , find all asymptotes.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
For each rational function , find all asymptotes. b.
Solution

36. As x gets larger and larger, both approach 0, and (x) approaches
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
As x gets larger and larger, both
approach 0, and (x) approaches
so the line y = 2 is the horizontal asymptote.

37. For each rational function , find all asymptotes.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
For each rational function , find all asymptotes. c.
Solution Setting the denominator x – 2 equal to 0 shows that the vertical asymptote has equation x = 2. If we divide by the greatest power of x as before ( in
this case), we see that there is no horizontal asymptote because

38. does not approach any real number as x  , since
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
does not approach any real number as x  , since
is undefined. This happens whenever the degree of the numerator is greater than the degree of the denominator. In such cases, divide the denominator into the numerator to write the expression in another form. We use synthetic division.

39. We use synthetic division.
FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
We use synthetic division.
The result allows us to write the function as

40. FINDING ASYMPTOTES OF GRAPHS OF RATIONAL FUNCTIONS
Example 4
For very large values of x, is close to 0, and the graph approaches the line
y = x + 2. This line is an oblique asymptote (slanted, neither vertical nor horizontal) for the graph of the function.

41. Steps for Graphing Functions
A comprehensive graph of a rational function exhibits these features:
1. all x- and y-intercepts;
2. all asymptotes: vertical, horizontal, and/or oblique;
3. the point at which the graph intersects its nonvertical asymptote (if there is any such point);
4. the behavior of the function on each domain interval determined by the vertical
asymptotes and x-intercepts.

42. Graphing a Rational Function
Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps.
Step 1 Find any vertical asymptotes.
Step 2 Find any horizontal or oblique asymptotes.
Step 3 Find the y-intercept by evaluating (0).

43. Graphing a Rational Function
Step 4 Find the x-intercepts, if any, by solving
(x) = 0 . (These will be the zeros of the numerator, p(x).)
Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving (x) = b or(x) = mx + b.

44. Graphing a Rational Function
Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts.
Step 7 Complete the sketch.

45. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 1 Since 2x2 + 5x – 3 = (2x – 1)(x + 3), from Example 4(a), the vertical asymptotes have equations x = ½ and x = – 3.
Step 2 Again, as shown in Example 4(a), the horizontal asymptote is the x-axis.

46. Step 3 The y-intercept is – ⅓, since
GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 3 The y-intercept is – ⅓, since
The y-intercept is the ratio of the constant terms.

47. Graph Solution Step 4 The x-intercept is found by solving (x) = 0.
GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 4 The x-intercept is found by solving (x) = 0.
If a rational expression is equal to 0, then its numerator must equal 0.
The x-intercept is – 1.

48. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 5 To determine whether the graph intersects its horizontal asymptote, solve
y-value of horizontal asymptote
Since the horizontal asymptote is the x-axis, the solution of this equation was found in Step 4. The graph intersects its horizontal asymptote at (– 1, 0).

49. GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 6 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each interval.

50. Example 5 (– , – 3) – 4 Negative Below (– 3, – 1) – 2 Positive Above
GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Interval
Test Point
Value of (x)
Sign of (x)
Graph Above or Below
x-Axis
(– , – 3)
– 4
Negative
Below
(– 3, – 1)
– 2
Positive
Above
(– 1, ½ )
(½ , ) 2

51. Step 7 Complete the sketch.
GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE
Example 5
Graph
Solution
Step 7 Complete the sketch.

52. GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE
Example 6
Graph
Solution
Step 1 and 2 As determined in Example 4(b), the equation of the vertical asymptote is x = 3. The horizontal asymptote has equation y = 2.

53. Step 3 (0) = – ⅓, so the y-intercept is – ⅓.
GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE
Example 6
Graph
Solution
Step 3 (0) = – ⅓, so the y-intercept is – ⅓.

54. Step 4 Solve (x) = 0 to find any x-intercepts.
GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE
Example 6
Graph
Solution
Step 4 Solve (x) = 0 to find any x-intercepts.
If a rational expression is equal to 0, then its numerator must equal 0.
x-intercept

55. GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE
Example 6
Graph
Solution
Step 5 The graph does not intersect its horizontal asymptote since (x) = 2 has no solution.

56. GRAPHING A RATIONAL FUNCTION THAT DOES NOT INTERSECT ITS HORIZONTAL ASYMPTOTE
Example 6
Graph
Solution
Step 6 and 7 The points (– 4, 1), (1, – 3/2), and (6, 13/3) are on the graph and can be used to complete the sketch.

57. Step 1 To find the vertical asymptote(s), solve x2 + 8x + 16 = 0.
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 1 To find the vertical asymptote(s), solve x2 + 8x + 16 = 0.
Set the denominator equal to 0.
Factor.
Zero-factor property.

58. GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Zero-factor property.
Since the numerator is not 0 when x = – 4, the vertical asymptote has the equation x = – 4.

59. GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 2 We divide all terms by x2 to get the equation of the horizontal asymptote.
Leading coefficient of numerator
Leading coefficient of denominator

60. Step 3 The y-intercept is (0) = – 3/8.
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 3 The y-intercept is (0) = – 3/8.

61. Step 4 To find the x-intercept(s), if any, we solve (x) = 0.
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 4 To find the x-intercept(s), if any, we solve (x) = 0.
Set the numerator equal to 0.

62. The x-intercepts are – 1 and 2.
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 4
Set the numerator equal to 0.
Divide by 3.
Factor.
Zero-factor property
The x-intercepts are – 1 and 2.

63. GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 5 We set (x) = 3 and solve to locate the point where the graph intersects the horizontal asymptote.
Multiply by x2 + 8x + 16.

64. Graph Solution Step 5 Example 7
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 5
Multiply by x2 + 8x + 16.
Subtract 3x2.
Subtract 24x; add 6.
Divide by – 27.

65. The graph intersects its horizontal asymptote at (– 2, 3).
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 5
Divide by – 27.
The graph intersects its horizontal asymptote at (– 2, 3).

66. Graph Solution Example 7
GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE
Example 7
Graph
Solution
Step 6 and 7 Some of the other points that lie on the graph are
These are used to complete the graph.

67. Behavior of Graphs of Rational Functions Near Vertical Asymptotes
Suppose that (x) is defined by a rational expression in lowest terms. If n is the largest positive integer such that (x – a)n is a factor of the denominator of (x), the graph will behave in the manner illustrated.

68. Behavior of Graphs
In Section 3.4 we observed that the behavior of the graph of a polynomial function near its zeros is dependent on the multiplicity of the zero. The same statement can be made for rational functions. Suppose that (x) is defined by a rational expression in lowest terms. If n is the greatest positive integer such that (x – c)n is a factor of the numerator of (x), the graph will behave in the manner illustrated.

69. Behavior of Graphs

70. GRAPHING A RATIONAL
FUNCTION WITH AN OBLIQUE ASYMPTOTE
Example 8
Graph
Solution In Example 4, the vertical asymptote has equation x = 2, and the graph has an oblique asymptote with equation y = x Refer to the previous discussion to determine the behavior near the vertical asymptote x = 2.

71. GRAPHING A RATIONAL
FUNCTION WITH AN OBLIQUE ASYMPTOTE
Example 8
Graph
Solution The y-intercept is – ½ , and the graph has no x-intercepts since the numerator, x2 + 1, has no real zeros. The graph does not intersect its oblique asymptote because

72. has no solution. Using the y-intercept, asymptotes, the points and
GRAPHING A RATIONAL
FUNCTION WITH AN OBLIQUE ASYMPTOTE
Example 8
Graph
Solution
has no solution. Using the y-intercept, asymptotes, the points and
and the general behavior of the graph near its asymptotes leads to this graph.

73. Graph Solution Example 8 GRAPHING A RATIONAL
FUNCTION WITH AN OBLIQUE ASYMPTOTE
Example 8
Graph
Solution

74. Graph Solution Example 8
GRAPHING A RATIONAL
FUNCTION WITH AN OBLIQUE ASYMPTOTE
Example 8
Graph
Solution
As mentioned earlier, a rational function must be defined by an expression in lowest terms before we can use the methods discussed in this section to determine the graph. A rational function that is not in lowest terms usually has a “hole,” or point of discontinuity, in its graph.

75. Solution The domain of this function cannot
GRAPHING A RATIONAL FUNCTION DEFINED BY AN EXPRESSION THAT IS NOT IN LOWEST TERMS
Example 9
Graph
Solution The domain of this function cannot
include 2. The expression should be written in lowest terms.
Factor.

76. GRAPHING A RATIONAL FUNCTION DEFINED BY AN EXPRESSION THAT IS NOT IN LOWEST TERMS
Example 9
Graph
Solution The graph of this function will be the same as the graph of
y = x + 2 (a straight line), with the exception of the point with x-value 2. A “hole” appears in the graph at (2, 4).

77. MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION
Example 10
Vehicles arrive randomly at a parking ramp at an average rate of 2.6 vehicles per minute. The parking attendant can admit 3.2 vehicles per minute. However, since arrivals are random, lines form at various times.

78. MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION
Example 10
(a) The traffic intensity x is defined as the ratio of the average arrival rate to the average admittance rate. Determine x for this parking ramp.
Solution
The average arrival rate is 2.6 vehicles and the average admittance rate is
3.2 vehicles, so

79. MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION
Example 10
(b) The average number of vehicles waiting in line to enter the ramp is given by
where 0  x < 1 is the traffic intensity. Graph (x) and compute (.8125) for this parking ramp.
Solution

80. MODELING TRAFFIC INTENSITY WITH A RATIONAL FUNCTION
Example 10
(c) What happens to the number of vehicles waiting as the traffic intensity approaches 1?
Solution
From the graph we see that as x approaches 1, y = (x) gets very large; that is, the average number of waiting vehicles gets very large. This is what we would expect.