Lesson 7.1 Skills Practice
Name
Date
A Rational Existence
Introduction to Rational Functions Vocabulary Write the term that best completes each sentence. 1. A
is any function that can be written as the ratio of two polynomials.
2. A never intersects.
is a vertical line that a function gets closer and closer to, but
Problem Set Determine whether each function is a rational function or not a rational function. If the function is not rational, explain why. 1. f(x) 5 x2 2 6x 1 2
2
2. g(x) 5 x3 2 1
© Carnegie Learning
The function f(x) is a rational function.
( )
3 1 1 __ 3. q(x) 5 ___________ x 2 2x √ x
x 4. r(x) 5 __ 1 3
5. h(x) 5 _____ x x23
6. t(x) 5 __ 5 x 2 x
7. s(x) 5 1 2 4x
1 8. p(x) 5 ____________ (x 2 1)(x 1 1)
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Lesson 7.1 Skills Practice
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Describe the vertical and horizontal asymptotes for each graph, provided they exist. Each figure represents the graph of a rational function. y
9.
y
10.
1
1 0
21
1
x
0
21
21
1
x
1
x
21
The vertical asymptote is the y-axis or x 5 0. The horizontal asymptote is the x-axis or y 5 0.
y
11.
y
12.
1
1 0
21
1
0
21 21
© Carnegie Learning
21
x
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Lesson 7.1 Skills Practice
page 3
Name
Date
y
13.
y
14.
1
1 0
21 21
1
x
0
21
1
x
21
a . Note that a is a non-zero Describe the domain and range of each rational function of the form f(x) 5 __ xn real number and n is an integer greater than or equal to 1.
© Carnegie Learning
2 15. f(x) 5 __ x4
The domain of f(x) is the set of real numbers excluding 0. The range of f(x) is the set of real numbers greater than 0.
16. f(x) 5 ___ 21 x3
17. f(x) 5 __ 23 x
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Lesson 7.1 Skills Practice
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y
18.
1 0
21
1
x
1
x
1
x
21
19.
y
1 0
21 21
20.
1 0
21 21
© Carnegie Learning
y
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Lesson 7.1 Skills Practice
Name
page 5
Date
Describe the end behavior of each rational function of the form f(x) 5 __ an . Note that a is a non-zero real x number and n is an integer greater than or equal to 1. 21 21. f(x) 5 ___ x4
As x approaches negative infinity, y approaches 0. As x approaches positive infinity, y approaches 0.
22. f(x) 5 __ 13 x
23. f(x) 5 ___ 22 x3
y
24.
© Carnegie Learning
1 0
21
1
x
21
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Lesson 7.1 Skills Practice
page 6
y
25.
1 0
21
1
x
1
x
21
26.
y 1 0
21 21
Describe the behavior of each rational function as x approaches zero from the left and as x approaches zero from the right. Each rational function is in the form f(x) 5 __ an . Note that a is a non-zero real number x and n is an integer greater than or equal to 1.
x
2 2 __ 3 ____ 28. f(x) 5 3 x
As x approaches zero from the left, the y values approach infinity. As x approaches zero from the right, the y values approach infinity.
© Carnegie Learning
__ 1 __ 27. f(x) 5 24
29. f(x) 5 __ 15 x
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Lesson 7.1 Skills Practice
Name
page 7
Date
30.
y
1 0
21
1
x
1
x
21
y
31.
1 0
21
© Carnegie Learning
21
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Lesson 7.1 Skills Practice 32.
page 8
y
1 0
21
1
x
21
1 where n is an integer greater Analyze each key characteristic of a rational function of the form f(x) 5 __ xn than or equal to 1. Identify whether the given characteristic is modeled by an odd power of n, an even power of n, or both. 33. Range is all real numbers excluding 0.
34. Domain is all real numbers excluding 0.
35. Horizontal asymptote at x 5 0.
36. Graph only exists in the first and second quadrants.
37. Graph could be in any of the quadrants.
38. Range is all real numbers greater than 0.
© Carnegie Learning
This characteristic is modeled by an odd power of n.
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Lesson 7.2 Skills Practice
Name
Date
A Rational Shift in Behavior Translating Rational Functions Problem Set
© Carnegie Learning
Complete the table. Use your graphing calculator to help.
c-value
1 g(x) 5 ______ x 2 c
Vertical Asymptote(s)
Horizontal Asymptote(s)
Domain
Range
1.
3
1 g(x) 5 ______ x23
x53
y50
Real Numbers except 3
Real Numbers except 0
2.
24
3.
__ 1
4.
22.7
5.
5
6.
7 2 __ 8
2
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Lesson 7.2 Skills Practice
page 2
Determine the domain, range, and vertical and horizontal asymptotes of each rational function without using a graphing calculator. 7. f (x) 5 __ 3 x
8. f (x) 5 _____ 1 x29
Domain: All real numbers except 0. Range: All real numbers except 0. Vertical Asymptote at x 5 0. Horizontal Asymptote at y 5 0.
9. f (x) 5 _____ 1 x19
10. f (x) 5 2x 1 1
11. f(x) 5 _______ 27 2x 1 3
x 12. f (x) 5 __ 7
13. Vertical asymptote at x 5 2 and a horizontal asymptote at y 5 0. Answers will vary. 1 f(x) 5 ______ x22 The denominator cannot be 2, so there will be a vertical asymptote at x 5 2. The function has a constant in the numerator and variable in the denominator, so the output will approach 0 as x increases or decreases, creating a horizontal asymptote at y 5 0.
© Carnegie Learning
Write a rational function for each table, graph, or description provided. Explain your reasoning.
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Lesson 7.2 Skills Practice
Name
page 3
Date
14. Vertical asymptote at x 5 25 and a horizontal asymptote at y 5 0.
15.
x
f(x)
24
21
23
3 2 __ 2
22
23
21
undefined
0
3
1
__ 3
2
1
2
© Carnegie Learning
16. The domain is all real numbers except x 5 6. The range is all real numbers except y 5 0.
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Lesson 7.2 Skills Practice
page 4
17. Vertical asymptote at x 5 23. The range is all real numbers except y 5 0.
18.
y
1
0
x
1
21 21
Sketch each rational function without using a graphing calculator. 1 20. f(x) 5 __ 12 19. f(x) 5 __ x x
1
1 0
21
1
21
x
0
21 21
1
x
© Carnegie Learning
y
y
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Lesson 7.2 Skills Practice
page 5
Name
Date
21. f(x) 5 _____ 1 x24
22. f(x) 5 _____ 21 x24 y
1
y
1
0
21 21
x
1
23. f(x) 5 _______ 1 2 (x 2 4)
1
x
24. f(x) 5 _______ 21 2 (x 2 4) y
y
1 21 0 1 21
© Carnegie Learning
0
21 21
1 x
21 0 1 21
x
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Lesson 7.2 Skills Practice
page 6
Analyze each rational function. Use algebra to determine the vertical asymptotes. Do not use a graphing calculator. 25. f(x) 5 ________ 2 3x 2 15
1 26. f(x) 5 ____________ (x 1 2)(x 2 3)
3x 2 15 5 0 3x 5 15 x 5 5 A vertical asymptote exists at x 5 5.
28. f(x) 5 ______ 2 x x 2x
7 29. f(x) 5 ______ x2 1 1
30. f(x) 5 ____________ x 2 3 (x 2 3)(x 2 2)
© Carnegie Learning
12x 27. f(x) 5 ___________ x2 1 4x 2 5
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Lesson 7.2 Skills Practice
Name
page 7
Date
Determine two different rational functions with the given characteristics. 31. The rational functions have a vertical asymptote at x 5 21. Answers will vary. f(x) 5 5 or g(x) 5 21 2 x11 (x 1 1)
______
_______
32. The rational functions have a vertical asymptote at x 5 0.
33. The rational functions have vertical asymptotes at x 5 24 and x 5 2.
34. The rational functions have vertical asymptotes at x 5 0 and x 5 7.
© Carnegie Learning
35. The rational functions have a vertical asymptote at x 5 3 and a y-intercept of (0, 21).
36. The rational functions have a vertical asymptote x 5 22. Also they each have a second vertical asymptote but not the same one.
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© Carnegie Learning
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Lesson 7.3 Skills Practice
Name
Date
A Rational Approach Exploring Rational Functions Graphically Problem Set Sketch each function without using a graphing calculator. Indicate the domain, range, vertical and horizontal asymptote(s), and y-intercept. 1. f(x) 5 _____ 1 x23 Domain: All real numbers except 3.
y
Range: All real numbers except 0. Asymptote(s): Vertical asymptote at x 5 3. Horizontal asymptote at y 5 0. 0
x
(
)
1 y-intercept: 0, 2 __ 3
1 2. f(x) 5 ____________ (x 1 2)(x 2 4) Domain:
y
© Carnegie Learning
Range: Asymptote(s):
0
x
y-intercept:
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Lesson 7.3 Skills Practice
page 2
3. f(x) 5 _______ 2 1 x 2 3x Domain:
y
Range: Asymptote(s):
0
x
y-intercept:
4. f(x) 5 ___________ 2 1 x 1 x 2 6 Domain:
y
Range: Asymptote(s):
0
x
y-intercept:
© Carnegie Learning
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Lesson 7.3 Skills Practice
page 3
Name
Date
5. f(x) 5 ______ 2 1 x 21 Domain:
y
Range: Asymptote(s): 0
x
y-intercept:
1 6. f(x) 5 ___________ 2 x 1 4x 1 4 Domain:
y
Range: Asymptote(s):
© Carnegie Learning
0
x
y-intercept:
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Lesson 7.3 Skills Practice
page 4
The function f(x) 5 __ 1 x is shown on each coordinate plane. Determine whether the other function shown is the graph of g(x), p(x), or q(x). Explain your reasoning. 7. g(x) 5 _____ 1 x23 1 _____ p(x) 5 x13 q(x) 5 __ 1 x 1 3 Function: g(x) 5 ______ 1 x23
y
Explanation: The original function f(x) 5 __ 1 x has been shifted 3 units to the right. This results from a change in the C value. 0
x
1 8. g(x) 5 _____ x23 p(x) 5 _____ 1 x13 __ q(x) 5 1 x 1 3 Function:
y
0
x
© Carnegie Learning
Explanation:
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Lesson 7.3 Skills Practice
page 5
Name
Date
9. g(x) 5 _____ 1 x24 p(x) 5 _____ 1 x14 q(x) 5 __ 1 x 2 4 Function:
y
Explanation:
0
x
1 10. g(x) 5 _____ x24 p(x) 5 _____ 1 x14 q(x) 5 __ 1 x 2 4 Function:
y
© Carnegie Learning
Explanation:
0
x
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Lesson 7.3 Skills Practice
page 6
11. g(x) 5 __ 3 x p(x) 5 ___ 23 x
q(x) 5 __ 3 x 1 2 Function:
y
Explanation:
0
x
3 12. g(x) 5 __ x p(x) 5 ___ 23 x
q(x) 5 __ 3 x 1 2 Function:
y
0
x
© Carnegie Learning
Explanation:
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Lesson 7.3 Skills Practice
page 7
Name
Date
Sketch g(x) on each coordinate plane, given f(x) 5 __ 1 x . 14. g(x) 5 f (x) 2 4
13. g(x) 5 f (x 2 2) y
y
0
0
x
x
g(x) 1 x 2
15. g(x) 5 f (x 1 3)
16. g(x) 5 2f (x) y
y
x
0
x
© Carnegie Learning
0
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Lesson 7.3 Skills Practice
page 8 18. g(x) 5 f (x 2 1) 1 1
17. g(x) 5 f (x 1 2) 2 1 y
0
y
0
x
x
Write a rational function g(x) that matches the given characteristic(s). 19. Vertical asymptote at x 5 5
Answers will vary. g(x) 5 ______ 1 x25
20. Vertical asymptotes at x 5 22 and x 5 1
21. Vertical asymptote at x 5 4 Horizontal asymptote at y 5 23
22. Vertical asymptotes at x 5 23 and x 5 5
23. For f(x) 5 __ 1 x , g(x) 5 f(x 1 7) 2 2.
24. For f(x) 5 __ 1 x , g(x) shifts f(x) left 1 unit and down 2 units.
© Carnegie Learning
Horizontal asymptote at y 5 1
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Lesson 7.4 Skills Practice
Name
Date
There’s a Hole In My Function, Dear Liza Graphical Discontinuities Vocabulary Write a definition for the term in your own words. 1. removable discontinuity
Problem Set Determine which function, f(x) or g(x), has a removable discontinuity without using your graphing calculator. Identify the removable discontinuity. 1. f(x) 5 _____ 1 x22
2. f(x) 5 _______ 1 x(x 1 6)
x 1 4 g(x) 5 ____________ (x 2 3)(x 1 4)
x(x 1 1) g(x) 5 _______ x
The function g(x) has a removable discontinuity at x 5 24.
© Carnegie Learning
x 1 2 3. f(x) 5 _______ (x 1 2)2
3 4. f(x) 5 __ xx
g(x) 5 (x 2 3)(x 1 7)
2 g(x) 5 __ x
5. f(x) 5 _____ x 1 1 x23 (x 1 4)(x 1 7) g(x) 5 __________________ (x 2 2)(x 2 1)(x 1 7)
x2 (x 2 1) 6. f(x) 5 ________ x(x 2 3) g(x) 5 _______ x (x 2 1)
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Lesson 7.4 Skills Practice
page 2
Simplify each rational expression. List any restrictions on the domain. 7. _______ 3x 2 9 x23
3 (x 2 3) _______ 3x 2 9 5 ________ x23 x23 1
2xy 2 2y 8. _________ x21
1
x2 2 1 9. ______ x21
x 2 5 10. _______ x2 2 25
2 20 11. ___________ x 1 x 2 x15
x2 1 5x 2 14 12. ____________ x2 1 8x 1 7
3 13. ______ x 2 1 x21
2 2 8 14. ____________ x2 2 2x x 1 8x 1 15
© Carnegie Learning
5 3; x fi 3
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Lesson 7.4 Skills Practice
page 3
Name
Date
Determine whether the graph of each rational function has a vertical asymptote, a removable discontinuity, both, or neither. List the discontinuities, if any exist. 16. f(x) 5 _____ 2x x27
5 17. f(x) 5 ______ x 2 10
18. f(x) 5 ____________ x 2 4 (x 2 2)(x 2 4)
2 19. f(x) 5 _______ x 2 2 3x x 29
20. f(x) 5 ____________ 2 x 1 2 x 2 6x 2 16
© Carnegie Learning
x(x 1 3) 15. f(x) 5 _______ x13 The function f(x) has a removable discontinuity at x 5 23.
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Lesson 7.4 Skills Practice 3 2 2 1 21. f(x) 5 ______________ x 2 x 1 x x21
page 4 x 1 2 22. f(x) 5 _______ x4 1 2x2 2
Write an example of a rational function that models each of the given characteristics. 23. A vertical asymptote at x 5 27.
24. A removable discontinuity at x 5 8.
Answers will vary. f(x) 5 1 x17
25. A vertical asymptote at x 5 0.
26. A vertical asymptote at x 5 23 and x 5 5.
27. A vertical asymptote at x 5 3.
28. No vertical asymptote.
© Carnegie Learning
______
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Lesson 7.4 Skills Practice
page 5
Name
Date
Sketch each rational function without using a graphing calculator. Identify any restrictions. 29. f(x) 5 _______ 2x 1 2 x11
_______ 2(x 1 1) 5 ________
f(x) 5 2x 1 2 x11
y
1
x11 1
(21, 2)
5 2; x fi 21 0
x
30. f(x) 5 _______ 2 x x 2 4x y
x
© Carnegie Learning
0
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Lesson 7.4 Skills Practice
page 6
31. f(x) 5 ____________ 2 x 1 3 x 1 7x 1 12 y
0
x
2 10 32. f(x) 5 ____________ x 2 3x 2 x25
y
x
© Carnegie Learning
0
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Lesson 7.4 Skills Practice
Name
page 7
Date
2 14 33. f(x) 5 ____________ x2 2 5x 2 x 2 5x 2 14
y
0
x
2 6 34. f(x) 5 ___________ x 2 5x 2 x26
y
x
© Carnegie Learning
0
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Lesson 7.4 Skills Practice
page 8
2 35. f(x) 5 ______ 422 x x 24
y
0
x
3 6 36. f(x) 5 ___________ x 2 7x 2 x11 y
x
© Carnegie Learning
0
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Lesson 7.5 Skills Practice
Name
Date
The Breaking Point Using Rational Functions to Solve Problems Problem Set Solve each problem. Explain your reasoning. 1. Courtney plays softball. Her goal for the season is to have an overall batting average of 0.300 or better. Currently she has 45 base hits in 150 at bats. How many consecutive base hits must she get to reach her goal? Courtney must get at least 8 consecutive hits to reach her goal. The ratio of her hits to her at bats
____ ________
is 40 . Additional hits increases the numerator as well as the denominator, represented by the 150 ratio 40 1 x . The result of graphing the functions y 5 40 1 x and y 5 0.300 and finding their 150 1 x 150 1 x
________
point of intersection provides the solution.
© Carnegie Learning
2. Tito is mixing green and red paint. Currently his mixture is 3 parts green to 5 parts red. What is the least amount of red paint Tito needs to add so that the mixture is in the ratio 1 part green to 6 parts red?
3. Talk-Tell is a cellular service provider. They advertise that you can buy a monthly plan for as low as $75 per month as long as you buy a cell phone costing $200. If you buy the monthly plan along with the phone, how many months will it take for your average cost of owning the phone and the plan to be less than $90?
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Lesson 7.5 Skills Practice
page 2
4. Mr. Motley is tweaking the final exam he intends to give to his Algebra 2 students. The current test has 10 multiple choice and 7 free response questions. He would like that ratio of multiple choice to free response questions to be 2:3. How many free response questions does Mr. Motley need to add to his test to achieve the desired ratio?
5. Ms. Greenery owns a lawn service company. She placed the following flyer in the mailboxes of everyone living in the town of Stork. Greenery Lawn Service • One-time fee $150 • Monthly application fee $25
6. Conroy is a budding entomologist, that means that he likes to study insects. In fact, Conroy has an insect collection that currently contains 30 insects that fly and 45 insects that crawl. He would like his collection to contain enough insects so that the ratio of the number of insects that fly to the number of insects that crawl is 1:2. How many insects that crawl should Conroy add to his collection to achieve the desired ratio?
© Carnegie Learning
If you purchase Ms. Greenery’s service, how many months will it take for your average monthly cost for the service to be less than $75?
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Lesson 7.5 Skills Practice
page 3
Name
Date
7. Upon entering Sly’s Long Term Rentals one is greeted by the following two signs. Big Deal 600 HD Television • $300 Deposit • $80 Monthly Rental Fee
Not So Big Deal 600 HD Television • $100 Deposit • $120 Monthly Rental Fee
Determine the number of months for which the average monthly cost for the Not So Big Deal is better than the average monthly cost for the Big Deal.
© Carnegie Learning
8. Milton and Tory both work at Widget Kingdom, a company that produces widgets. Milton is paid $112 a day plus $0.10 for each widget he produces; while Tory is paid $96 a day plus $0.15 for each widget she produces. If Milton and Tory consistently produce the same number of widgets, at what point will the average cost of a widget produced by Tory be greater than the average cost of a widget produced by Milton?
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Lesson 7.5 Skills Practice
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9. Cody and Melissa play on the same basketball team. Up to this point in the season Cody has made 19 out of 30 free throw attempts for an average of about 63.33%; while Melissa has made 10 out of 25 free throw attempts for an average of 40%. Suppose that during the balance of the basketball season Melissa shoots twice as many free throws as Cody and both Melissa and Cody make all of them. If at the end of the season they both end up with the same free throw average, what is the least number of free throws made by Cody during the balance of the season?
© Carnegie Learning
10. Condoleezza works from home 6 hours per day 5 days a week. The company she works for pays her $22 per hour plus $5 for every prospective customer she contacts who signs up for the service provided by the company. The company will continue to employ Condoleezza provided she can maintain an average weekly cost per new customer that is less than or equal to $7. How many new customers per week must Condoleezza sign up for the company’s service to be able to keep her job?
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Lesson 7.5 Skills Practice
page 5
Name
Date
Sketch a graph to solve each equation. Do not use a graphing calculator. 1 5 0 52 12. _____ x 1 11. _____ 2 x x13 y
y
(22, 2) 1
1 0
21
1
x
0
21
21
1
x
21
x 5 22
54 14. _____ 5x x21
5 2 13. _____ x 2 2 x23
y
y
© Carnegie Learning
1
1
0
21 21
1
x
1 21 0 21
x
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Lesson 7.5 Skills Practice 2 5 22 15. ______ x 2 9 x23
page 6 56 16. _____ 2x x12
y
y
1
1 21 0 21
1
1 21 0 21
x
x
© Carnegie Learning
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