171S4.5q Rational Functions
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 4: Polynomial and Rational Functions
4.1 Polynomial Functions and Models
4.2 Graphing Polynomial Functions
4.3 Polynomial Division; The Remainder and Factor Theorems
4.4 Theorems about Zeros of Polynomial Functions
4.5 Rational Functions
4.6 Polynomial and Rational Inequalities
Graphing rational functions (6 minutes) http://www.youtube.com/watch?v=YyNNdBv66VA
Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Special Functions. Enter the User ID from your instructor and the Quick Launch # 1436 (Grapher: Rational Functions ­ explore graphs)
March 31, 2013
4.5 Rational Functions
• For a rational function, find the domain and graph the function, identifying all of the asymptotes.
• Solve applied problems involving rational functions.
A rational function is a function f that is a quotient of two polynomials, that is, where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x) ≠ 0.
Example
Consider .
Find the domain and graph f.
The graph of the function is the graph of y = 1/x translated to the left 4 units.
Solution: When the denominator x + 4 = 0, we have x = ­4, so the only input that results in a denominator of 0 is ­4. Thus the domain is {x|x ≠ ­4} or Click the globe to the left and view the 21­minute Youtube video from Khan Academy that presents an excellent discussion and explanation of Asymptotes of Rational Functions.
(­∞, ­4) U (­4, ∞).
Oct 27­1:04 PM
Vertical Asymptotes
The vertical asymptotes of a rational function f(x) = p(x)/q(x) are found by determining the zeros of q(x) that are not also zeros of p(x). If p(x) and q(x) are polynomials with no common factors other than constants, we need to determine only the zeros of the denominator q(x).
If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote for the graph of the function.
See Interactive Figures in Multimedia Library of MyLab/
Mastering (CourseCompass). Section 4.5 Graphs of Rational Functions: Part 1 and Section 4.5 Graphs of Rational Functions: Oblique Asymptotes
Oct 27­1:04 PM
Horizontal Asymptote
The line y = b is a horizontal asymptote for the graph of f if either or both of the following are true:
When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.
Example: Find the horizontal asymptote: The numerator and denominator have the same degree. The ratio of the leading coefficients is 6/9, so the line y = 2/3 is the horizontal asymptote.
Determining a Horizontal Asymptote
Example
Determine the vertical asymptotes of the function.
(1) When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.
Factor to find the zeros of the denominator:
x2 ­ 4 = (x + 2)(x ­ 2)
(2) When the degree of the numerator of a rational function is less than the degree of the denominator, the x­axis, or y = 0, is the horizontal asymptote.
Thus, the vertical asymptotes are the lines x = ­2 and x = 2.
(3) When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote.
Oct 27­1:04 PM
Oct 27­1:04 PM
Example
True Statements
• The graph of a rational function never crosses a vertical asymptote.
Graph will not cross vertical asymptote.
f(x) = 2x / (x ­ 2)
When q(x) = 0, f(x) is undefined.
• The graph of a rational function might cross a horizontal asymptote but does not necessarily do so.
Graph Vertical asymptotes: x + 3 = 0, so x = ­3.
The degree of the numerator and denominator is the same. Thus, y = 2 is the horizontal asymptote.
1. Draw the asymptotes with dashed lines.
2. Compute and plot some ordered pairs and draw the curve.
Graph may cross horizontal asymptote.
f(x) = 5x / (x2 + 1)
• The graph of a rational function might cross a slant asymptote but does not necessarily do so.
Graph may cross slant asymptote.
f(x) = x3 / (x2 + 2)
Oct 27­1:04 PM
Oct 27­1:04 PM
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171S4.5q Rational Functions
Oblique or Slant Asymptote
Find all the asymptotes of .
The line x = ­2 is a vertical asymptote.
There is no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.
Note that
Divide to find an equivalent expression.
The line y = 2x − 1 is an oblique (or slant) asymptote.
March 31, 2013
Occurrence of Lines as Asymptotes of Rational Functions
For a rational function f(x) = p(x)/q(x), where p(x) and q(x) have no common factors other than constants:
(1) Vertical asymptotes occur at any x­values that make the denominator = 0.
(2) The x­axis is the horizontal asymptote when the degree of the numerator is less than the degree of the denominator.
(3) A horizontal asymptote other than the x­axis occurs when the numerator and the denominator have the same degree.
(4) An oblique (or slant) asymptote occurs when the degree of the numerator is exactly 1 greater than the degree of the denominator.
(5) There can be only one horizontal asymptote or one oblique asymptote and never both.
(6) An asymptote is not part of the graph of the function.
Graphing Rational Functions
1. Find the real zeros of the denominator. Determine the domain of the function and sketch any vertical asymptotes.
2. Find the horizontal or oblique asymptote, if there is one, and sketch it.
3. Find the zeros of the function. The zeros are found by determining the zeros of the numerator. These are the first coordinates of the x­intercepts of the graph.
4. Find f (0). This gives the y­intercept (0, f (0)), of the function.
5. Find other function values to determine the general shape. Then draw the graph.
Oct 27­1:04 PM
Example
Oct 27­1:04 PM
More Examples
Graph Graph the following functions.
1. Find the zeros by solving: a) The zeros are ­1/2 and 3, thus the domain excludes these values.
The graph has vertical asymptotes at x = 3 and x = ­1/2. We sketch these with dashed lines.
2. Because the degree of the numerator is less than the degree of the denominator, the x­axis, y = 0, is the horizontal asymptote.
b) 3. To find the zeros of the numerator, we solve x + 3 = 0 and get x = ­3. Thus, ­3 is the zero of the function, and the pair (­3, 0) is the x­intercept.
4. We find f(0):
Thus (0, ­1) is the y­intercept. 5. We find other function values to determine the general shape of the graph and then draw the graph.
Oct 27­1:04 PM
c) Graph (a)
1. Vertical Asymptote
x = ­2
2. Horizontal Asymptote
y = 1
3. x­intercept
(3, 0)
4. y­intercept
(0, ­3/2)
Oct 27­1:04 PM
Summary ASYMPTOTES OF RATIONAL FUNCTIONS
Graph (b)
1. Vertical Asymptote
x = ­3, x = 3
2. Horizontal Asymptote
y = ­1
3. x­intercepts
(±2.828, 0)
4. y­intercept
(0, ­8/9)
Consider the leading terms of the rational function.
Vertical Asymptote (V.A.):
If q(x) = 0, x = a is VERTICAL asymptote. Horizontal Asymptote (H.A.):
(1) If n < m, y = 0 is H.A.
Graph (c)
(2) If n = m, is H.A.
1. Vertical Asymptote
x = ­1
2. Oblique Asymptote
y = x ­1
3. x­intercept
(0, 0)
4. y­intercept
(0, 0)
Oct 27­1:04 PM
(3) If n > m, there is NO H.A.
Slant (Oblique) Asymptote (S.A.):
If n = m + 1, then slant asymptote is y = quotient when p(x) is divided by q(x) using long division.
Oct 28­6:58 AM
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171S4.5q Rational Functions
March 31, 2013
4.5 Rational Functions
362/2. Determine the domain of function.
f(x) = 1 / x3
362/4. Determine the domain of function.
f(x) = (x + 4)2 / (4x ­ 3)
Oct 31­10:03 AM
4.5 Rational Functions
362/5. Determine the domain of function.
f(x) = (3x ­ 4) / (3x + 15)
Mar 26­7:47 AM
4.5 Rational Functions
362/8. List all asymptotes and match with the appropriate graph.
f(x) = 8 / (x2 + 4)
362/6. Determine the domain of function.
f(x) = (x2 + 3x ­ 10) / (x2 + 2x)
362/10. List all asymptotes and match with the appropriate graph.
f(x) = 8x2 / (x2 ­ 4)
Mar 26­7:47 AM
Mar 26­7:47 AM
4.5 Rational Functions
4.5 Rational Functions
362/11. List all asymptotes and match with the appropriate graph.
362/14. Determine the vertical asymptote of the graph of the function f(x) = 4x / (x2 + 10x)
f(x) = 8x / (x ­ 4)
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2
362/12. List all asymptotes and match with the appropriate graph.
f(x) = 8x3 / (x2 + 4)
362/16. Determine the vertical asymptote of the graph of the function f(x) = (x4 + 2) / x
Mar 26­7:47 AM
Mar 26­7:47 AM
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171S4.5q Rational Functions
4.5 Rational Functions
362/18. Determine the vertical asymptote of the graph of the function f(x) = (x2 ­ 4) / x(x + 5)(x ­ 2)
362/20. Determine the vertical asymptote of the graph of the function f(x) = (x + 5) / (x2 + 4x ­ 32)
Mar 26­7:47 AM
4.5 Rational Functions
362/25. Determine the horizontal asymptote of the graph of the function f(x) = (x3 ­ 2x2 + x ­ 1) / (x2 ­ 16)
362/26. Determine the horizontal asymptote of the graph of the function f(x) = (8x4 + x ­ 2) / (2x4 ­ 10)
Mar 26­7:47 AM
March 31, 2013
4.5 Rational Functions
362/22. Determine the horizontal asymptote of the graph of the function g(x) = (x + 6) / (x3 + 2x2)
362/24. Determine the horizontal asymptote of the graph of the function f(x) = x5 / (x5 + x)
Mar 26­7:47 AM
4.5 Rational Functions
362/28. Determine the oblique (or slant) asymptote of the graph of the function f(x) = (x2 ­ 6x) / (x ­ 5)
363/29. Determine the oblique (or slant) asymptote of the graph of the function f(x) = (x4 ­ 2) / (x3 + 1)
Mar 26­7:47 AM
4.5 Rational Functions
363/30. Determine the oblique (or slant) asymptote of the graph of the function f(x) = (12x3 ­ x) / (6x2 + 4)
363/32. Determine the oblique (or slant) asymptote of the graph of the function f(x) = (5x3 ­ x2 + x ­ 1) / (x2 ­ x + 2)
Mar 28­8:46 AM
Mar 26­7:47 AM
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171S4.5q Rational Functions
4.5 Rational Functions
363/48. List domain and x­ and y­intercepts and graph the function
f(x) = ­2 / (x ­ 3)2
March 31, 2013
4.5 Rational Functions
363/66. List domain and x­ and y­intercepts and graph the function
f(x) = (x + 2) / (x2 + 2x ­ 15)
363/68. List domain and x­ and y­intercepts and graph the function
f(x) = (2x2 ­ 3x ­ 9) / (x2 ­ 2x ­ 3)
363/52. List domain and x­ and y­intercepts and graph the function
f(x) = ­1 / (x2 + 2)
Mar 26­7:47 AM
Mar 26­7:47 AM
4.5 Rational Functions
363/79. Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible: vertical asymptotes x = ­4, x = 5
4.5 Rational Functions
364/81. Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible: vertical asymptotes x = ­4, x = 5; horizontal asymptote y = 3/2; x­intercept (­2, 0).
364/80. Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible: vertical asymptotes x = ­4, x = 5; x­intercept (­2, 0).
364/82. Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible: oblique (or slant) asymptote y = x ­ 1
Mar 26­7:47 AM
Mar 26­7:47 AM
4.5 Rational Functions
4.5 Rational Functions
364/83. Medical Dosage. The function 364/84. Average Cost. The average cost per DVD, in dollars, for a company to produce x fitness workout DVDs is given by the function A(x) = (2x + 100) / x, where x > 0. a) Graph the function on the interval (0, ∞) and complete the gives the body concentration N(t), in parts per million, of a certain dosage of medication after time t, in hours.
a) Graph the function on the interval [15, ∞) and complete the following:
N(t) _____ as t ∞. b) Explain the meaning of the answer to part (a) in terms of the application. Mar 26­7:47 AM
following: A(x) _______ as x ∞. b) Explain the meaning of the answer to part (a) in terms of the application.
Mar 26­7:47 AM
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171S4.5q Rational Functions
March 31, 2013
4.5 Rational Functions
4.5 Rational Functions
364/85. Population Growth. The population P, in thousands, of a senior community is given by where t is the time, in months.
a) Graph the function on the interval [0, ∞). b) Find the population at t = 0, 1, 3, and 8 months.
c) Find the horizontal asymptote of the graph and complete the following: P(t) _______ as t ∞.
364/86. Minimizing Surface Area. The Hold­It Container Co. is designing an open­top rectangular box, with a square base, that will hold 108 cubic centimeters.
a) Express the surface area S as a function of the length x of a side of the base. b) Use a graphing calculator to graph the function on the interval (0, ∞). c) Estimate the minimum surface area and the value of x that will yield it. d) Explain the meaning of the answer to part (c) in terms of the application.
e) Find the maximum population and the value of t that will yield it. Mar 26­7:47 AM
Mar 26­7:47 AM
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