math 160 college algebra mock exam 1littrell.riomath.com/pdfs/math160mockexam1.pdf · coverage:...

MATH 160 COLLEGE ALGEBRA MOCK EXAM 1

Coverage: chapters 1 and 2 in the RHC custom edition of Sullivan & Sullivan's College Algebra, Concepts Through Functions, 3rd ed. (11 sections total). In the stock edition of this text, the coverage is chapter F (Foundations), and Chapter 1 (Functions and Their Graphs). In addition there are some algebra review items.

The contents of the actual exam will reflect the homework assignments and the material covered in class. This mock examination is offered to you as is, for informational purposes only. There are no guarantees either expressed or implied. For the full disclaimer, refer to your course Web page.

Your actual Exam 1 will consist of thirty multiple-choice questions, worth three points each, of which you must do at least twenty-five. Practically speaking, you never want to leave any multiple-choice questions blank, because they are automatically marked wrong when you do. If you answer every one, even if you guess, you can still have a reasonable chance of guessing correctly, so it is possible you could still get points, but we're only holding you accountable for twenty-five of the multiple-choice questions. In addition, there will be five open-ended questions, which you will need to work out in the pages of your exam paper. The five open-ended questions will be worth five points each; your exam score will be computed as a fraction of 100 points. This mock exam is a little longer than your actual exam (OK, almost twice as long). You have to understand, picking problems for mock exams is like eating potato chips... it's hard to stop. But having a nice, long mock exam is a good thing, because it gives you a lot of opportunity to practice problems which have less context than the ones in the book. Trust me, this is a real issue for a lot of people. They use the chapter section context as a crutch without even noticing that this is happening. Naturally, we want you to do well on the first exam, and on all the rest.

Remember, regardless of where you are taking this exam, you must supply a Scantron SC882-E/N-E/E-LOVAS for the exam (there are three versions of this form, the SC882-E, the SC882-N-E, and the SC882-E-LOVAS; any of the three versions is fine); no other forms will be accepted for full credit. In addition, you will need to know your student ID number when you sit for the exam. The SC882 form has an ID number field which we will be using; if you don't know your ID number or enter it incorrectly, there will be a penalty to your score. If you are taking the exam via ProctorU, you do not need a Scantron form; refer to the syllabus for the necessary details. Regardless of where you take the exam, it would also be wise to bring a legal graphing calculator. Please recall, the recommended models are anything from the TI-83/84 series. Neither computer algebra system graphing calculators (e.g., TI-89, TI-nSpire CAS), cell-phone calculators, or smart watches (e.g., iWatch) are permitted during exams. Again, refer to the syllabus for additional detail. While you're taking the exam you may request to use a copy of the Algebra Review Card (you can see it for yourself under the "Tools For Success" tab in MyLab). Finally, you will be permitted to use a single sheet of notes, no larger than 8.5" by 11". Write on one or both sides as you wish, but anything which appears on the sheet must be handwritten.

You will have a maximum of two hours to complete your exam. The proctor will time-stamp your exam; if you exceed the allocated time, your score will be reduced proportionally. For example, if you exceeded the allocated time by 12 minutes (12 minutes/120 minutes = 10%), your score would be reduced by 10%. If you are taking the exam via ProctorU, it will only be available for 120 minutes; there is no need for the proctor to stamp anything. In any case, because the open-ended questions are more valuable than the multiple-choice questions, it may be wise to begin your exam by working on the open-ended questions FIRST.

Finally: occasionally the algorithms behind these mock exam questions will throw a bad number, resulting in an incorrect answer on the key. If you find any inconsistencies on the key, like an apparently correct answer marked as incorrect, etc., I would greatly appreciate a heads-up. In addition, many exam bank questions are written so the answers mesh well with table use, instead of graphing calculator use. For that reason, some answers that are correct may appear to differ slightly from the answer obtained correctly on a graphing calculator. This may appear to be a roundoff error, but typically if you round the answer obtained via the calculator correctly, to the same number of places, the answers will mesh. If necessary, please email me directly regarding such items.

Use the following to answer questions 1-10:

Questions 1-10 cover review topics from Intermediate Algebra/High School Algebra II. If you find these questions are not difficult, that is a good sign. Regardless, I would encourage you to consider checking out Sal Khan's channel on YouTube (search for Khan Academy) or some equivalent. Mathematics is a vertically-structured subject, and it can be difficult when you have forgotten basic elements that are widely used in a course like this one.

Revision Date: Spring 2018

1. What is the GCF that can be factored out from 4 39 6 3n nx x x ?a) 3 nx b) 3x c) 23x d) 3x e) None of these.

2. Factor: 2 225 20 4z za a a) 5 2 5 2za za d) 2

5 2z a

b) Nonfactorable e) 5 2 5 2z a z a

c) 25 2z a

3. Which of the following (if any) can be factored using the difference of squares pattern? Mark all which apply.a) 8 22 16x xb) 7 596 54x xc) 6 48 2x xd) 432 162x e) None of these.

4. Which of the following (if any) can be factored using the sum or difference of cubes pattern? Mark all which apply.a) 6 6x yb) 9 527 8x xc) 424 3x xd) 9 354 128x xe) None of these.

5. Factor completely: 9 1227 64x y

a) 3 4 6 3 4 83 4 9 12 16x y x x y y d) 3 4 6 3 4 83 4 9 12 16x y x x y y

b) 3 4 6 3 4 83 4 9 24 16x y x x y y e) None of these.

c) 3 4 3 3 4 43 4 9 24 16x y x x y y

6. Which of the following, if any, can be factored by grouping? Mark all which apply.a) 8 6 12 9ax az bx bz b) 10 2 15 3ax bx az bz c) 10 2 15 3ac bc ax bx d) 9 6 12 8ax bx ay by e) None of these.


7. Simplify completely, expressing your answer with positive exponents only: 2 2

1 1

x y

y x

a) 1

x y b)

2 2

y x

x y

c) x y

x y

d) x y

xy

e) None of these.

8. Rationalize the denominator and simplify completely: 3

3

x

x

a) 3x b) 3 2 3

3

x x x

x

c) 3 2 3

3

x x x

x

d) 3x e) None of these.

9. Rationalize the denominator and simplify your answer completely: 3

6

9x

a) 32 3x

x b)

32 3

3

x

x c)

3 22 3x

x d)

32 9

3

x

x e) None of these.

10. What is the remainder when we divide 4 3 226 21 23 4x x x x x ?

a) 13 b) –285 c) 11 d) 10 e) None of these.

11. Find the midpoint of the line segment with endpoints (1, 2) and (7, 4).a) (–6, –2) b) (4, 3) c) (–3, –1) d) (8, 6) e) None of these.

12. Find the distance between (–3, –2) and (1, 4).

a) 2 5 b) 10 c) 27 d) 2 13 e) None of these.

13. Find the distance between the points. Round to the nearest hundredth, if necessary: (1, –1), (–6, –2)a) 5.1 b) 4.47 c) 5.83 d) 7.07 e) 7.62


14. Calculate (to two decimal places) the perimeter of the triangle with the following vertices at points A, B, and C.

a) 22.22 b) 2.57 c) 12.56 d) 172.00 e) None of these.

15. Find x such that the distance between the point –4,0 and , 4x is 5.

a) –8,0x b) –7,0x c) –8, –1x d) –7, –1x e) –1,0x

16. Given3

4 1

xy

x

, use the algebraic tests to determine symmetry with respect to both axes and the

origin.a) x-axis, y-axis, and origin symmetry d) no symmetryb) y-axis symmetry only e) origin symmetry onlyc) x-axis symmetry only


17. Assuming that the graph shown has y-axis symmetry, sketch the complete graph.

a)

b)


c)

d)

e)


18. Find the x- and y-intercepts of the graph of the following equation. 12 + 9 10x y

a) x-int: 5

,06

; y-int: 10

0,9

d) x-int: 4

,03

; y-int: 10

0,9

b) x-int: 4

,03

; y-int: 3

0,4

e) x-int: 6

,05

; y-int: 4

0,3

c) x-int: 5

,06

; y-int: 3

0,4


Consider the graph below.

19. What is the slope of line A?

a) 3

4 b)

4

3 c)

3

4 d)

4

3 e) None of these.

20. What is the slope of line B?

a) 3

2 b)

2

3 c)

3

2 d)

2

3 e) None of these.

21. Which of the following (if any) correspond to line A, pictured above?a) 3 4 8x y b) 3 4 8x y c) 3 4 8x y d) 4 3 6x y e) None of these.

22. Which of the following (if any) correspond to line B, pictured above?a) 3 2 11x y b) 2 3 11x y c) 3 2 11x y d) 2 3 11x y e) None of these.


23. Write an equation of the line passing through (–8, –3) and perpendicular to y = 1

24

x . Write your

answer in standard form Ax + By = C, A ≥ 0.a) x – 4y = –20 b) x + 4y = –20 c) 4x + y = –35 d) 4x – y = –35 e) None of these.


Consider the two lines graphed below.

24. What is the equation of line A, expressed in standard form?a) 4 3 6x y b) 3 4 8x y c) 4 3 6x y d) 3 4 8x y e) None of these.

25. What are the coordinates of the x-intercept of line B?

a) 19

,08

b) 21

,08

c) 5

,02

d) 13

,05

e) None of these.

26. What is the equation of line B in slope-intercept form?

a) 4 8

3 3y x b)

4 10

3 3y x c)

3 10

4 3y x d)

4 11

3 3y x e) None of these.

27. Which of the following (if any) are true of lines A and B?a) The product of the slopes is 1 .b) The lines are perpendicular.c) The slopes of lines A and B are negative reciprocals of each other.d) If we make a triangle using these lines, with vertices at 0,2 , the point where the two lines

intersect, and the y-intercept of line B, the triangle is a right triangle.e) None of these.



Consider the circle graphed below.

28. Which of the following (if any) correspond to the circle above?

a) 2 23 2 16x y d) 2 2

3 2 16x y

b) 2 23 2 4x y e) None of these.

c) 2 23 2 4x y

29. Which of the following (if any) correspond to the circle above?a) 2 2 6 4 3 0x y x y d) 2 2 6 4 9 0x y x y b) 2 2 6 4 3 0x y x y e) None of these.

c) 2 2 6 4 9 0x y x y

30. Which of the following (if any) correspond to the x-intercepts of the circle pictured above?

a) 3 7 b) 3 2 2 c) 3 2 3 d) 3 2 3 e) None of these.

31. The point (–6, –4) lies on a circle centered at (3, 2). Find the equation of the circle in standard form.a) (x – 3)2 + (y – 2)2 = 5 d) (x + 3)2 + (y + 2)2 = 117b) (x + 6)2 + (y + 4)2 = 117 e) None of these.c) (x – 3)2 + (y – 2)2 = 117


32. Use a graphing utility to graph the function and determine whether the function is even, odd, or neither.

2 4f x x x a) Even

b) Neither even nor odd

c) Odd

d) Neither even nor odd

e) Even

33. Decide whether the function is even, odd, or neither.

3 5g x x x a) Even b) Neither even nor odd c) Odd


34. Find the domain of the function.

2( ) 36f w w a) w –6 or w 6 b) w 6 c) all real numbers d) w 0 e) –6 w 6


Consider the function graphed below.

For a series of videos which will help you solve the problems which follow, scan the QR codes:


35. What is the domain of f x ?

a) 2, 4 b) 2, 4 c) 4, 4 d) 4, 4 e) None of these.

36. What is the range of 1 2f x ?

a) 4, 2 b) 4, 2 c) 1,5 d) 2, 4 e) None of these.

37. Over which interval(s) is the function increasing?a) 2,0 2, 4 d) 4, 2 2,0 2,4 b) 2, 2 2, 4 e) None of these.

c) 4, 2 0, 2

38. Where does the function have relative extrema (i.e., local maxima and/or local minima)?a) 4, 2,2,4x b) 2,0,2x c) 2,2x d) 4, 2,0,2,4x e) None of these.

39.

Consider the graph above, which is clearly related to f x by translation, reflection, expansion, and/or

compression. Which, if any, of the following correspond to this graph?a) 2a x f x d) 2a x f x

b) 2a x f x e) None of these.

c) 2a x f x


40.


compression. Which, if any, of the following correspond to this graph?a) b x f x d) 1b x f x

b) b x f x e) None of these.

c) 2b x f x

41.


compression. Which, if any, of the following correspond to this graph?a) 2c x f x d) 2c x f x

b) 2c x f x e) None of these.

c) 2c x f x


42.


compression. Which, if any, of the following correspond to this graph?

a) 21 2

3d x f x d) 1 2d x f x

b) 21 2

3d x f x e) None of these.

c) 21 2

3d x f x

43.


compression. Which, if any, of the following correspond to this graph?

a) 1

2e x f x

d) 2e x f x

b) 2e x f x e) None of these.

c) 1

2e x f x


44. Which of the following graphs from above, if any, are examples of odd functions?a) c x b) f x c) e x d) b x e) None of these.

45. Suppose the amount of federal income tax ( )T x an individual owed in 2006 is given by

T

if 0 75500.100.15 7550 755.00 if 7550 30,650

0.25 30,650 4220.00 if 30,650 74,200( )

0.28 74,200 15,107.50 if 74,200 154,800

0.33 154,800 37,675.50 if 154,800 336,550

0.35 336,550 97,653.00 if 336

xxx x

x xx

x x

x x

x x

,550

where x is the adjusted gross income tax of the taxpayer. Find the income tax owed by an individual whose adjusted gross income was $44,050.a) $7972.00 b) $7570.00 c) $6230.00 d) $6281.00 e) $3350.00


Consider the function:

2

24 3

3

4 3 3

1 3

x x

f x x x

x x


46. Which of the following, if any, shows the graph of f x ?

a) d)

b) e) None of these.

c)

47. What is 1f ?

a) 3 b) 2 c) –7 d) 13 e) None of these.

48. What is 1f ?

a) 3 b) 2 c) 1 d) –2 e) None of these.



Consider the graphs below:

49. Which of the graphs given above (if any) will pass the vertical line test? Mark all which apply.a) graph ab) graph bc) graph cd) graph de) None of these.

50. Which of the functions given above (if any) will pass the horizontal line test? Mark all which apply.a) graph ab) graph bc) graph cd) graph de) None of these.



Consider the graph of the function f x below.


51. What is the domain of f x , expressed in interval notation?

a) 4,4 b) 4, 4 c) 4, 4 d) 4, 4 e) None of these.

52. What is the range of f x , expressed in interval notation?

a) 0, 4 b) 0,4 c) 0,4 d) 0, 4 e) None of these.


53. What is 0f ?

a) 3 b) 1 c) 1 d) 2 e) None of these.

54. What is 1f ?

a) 2 b) 1 c) 2 d) 1 e) None of these.

55. Solve 2f x .

a) 1x b) 1,2x c) 2x d) 3x e) None of these.



Consider the graph of the function g x , shown below.



56. Which of the following functions, if any, correspond to the graph of g x shown above?

a) 3 3g x x d) 3 3g x x

b) 3 3g x x e) None of these.

c) 3 3g x x

57. Consider the function f x graphed above. What is the domain of the function, if expressed in

interval notation?a) , 3 3, d) ,

b) , 3 3, e) None of these.

c) ,3 3,

58. Consider the function f x graphed above. What is the range of the function, if expressed in interval

notation?a) , 0 0, b) , 0 0, c) ,1 1, d) , e) None of these.



Consider the graph of the function f x , shown below.


59. Which of the following functions, if any, correspond to the graph of f x shown above?

a) 11f x

x b) 1

1f x

x

c) 1

1f xx

d) 1

1f x

x

e) None of these.


60. Consider the function f x graphed above. What is the domain of the function, if expressed in

interval notation?a) ,1 1, d) , 0 0,

b) ,1 1, e) None of these.

c) , 1 1,

61. Consider the function f x graphed above. What is the range of the function, if expressed in interval

notation?a) , b) , 0 0, c) ,1 1, d) , 0 0, e) None of these.

62. An open box is to be made from a square piece of cardboard having dimensions 28 inches by 28 inches by cutting out squares of area 2x from each corner as shown in the figure below. Express the volume V of the box as a function of x.

28 2x

28 2x

a) 2 3( ) 28 56 4V x x x x d) 2 3( ) 784 56 4V x x x x b) 2 3( ) 784 112 4V x x x x e) 2( ) 784 112 4V x x x c) 2 3( ) 28 2V x x x

63. An open box is to be made from a square piece of cardboard having dimensions 36 inches by 36 inches by cutting out squares of area 2x from each corner as shown in the figure below. If the volume of the box is given by 2 3( ) 1296 144 4 ,V x x x x state the domain of V .

36 2x

36 2xa) all real numbers b) 144 1296x c) 4 144x d) 0 36x e) 0 18x


64. A rock attached to a string is whirled horizontally about the origin in a counterclockwise circular path with radius 7.5 feet. When the string breaks, the rock travels on a linear path perpendicular to the radius OP and hits the wall which is 15y feet away. If the string breaks when the rock is at

4.5,6 ,P find the x-coordinate of the point at which the rock hits the wall.

a) –10.5 b) –8.5 c) –9.5 d) –6.5 e) –7.5

65. The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers 18.0 hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour.a) 54.0 hours b) 10.8 hours c) 20.0 hours d) 16.0 hours e) None of these.

66. Write the equation that expresses the relationship between the variables, and then use the given data to solve for the variation constant.

T varies jointly as r and s and 233T when 10r and 7.s

a) 233

490k b)

70

233k c)

2330

7k d)

1631

10k e)

233

70k

67. The speed of a bicycle gear, in revolutions per minute, is inversely proportional to the number of teeth on the gear. If a gear with 60 teeth has a speed of 40 revolutions per minute, what will be the speed of a gear with 64 teeth? Round your answer to the nearest revolution per minute.a) 42 revolutions per minute d) 38 revolutions per minuteb) 39 revolutions per minute e) 41 revolutions per minutec) 35 revolutions per minute


Open-ended questions

When you sit for your actual exam, there will be at least six open-ended questions similar to the following, of which you must choose exactly five questions to work out in the space provided on your exam paper. You need to make sure to show all your work for these questions; I'm looking for solutions that are readable, easy to follow, and well-written... just like the examples in your text and/or the problems you've seen solved in the student solution manual.

I have a simple rule I apply when grading the solutions to these questions: no work, no credit. The work must appear in the space provided on your exam paper, and nowhere else. In other words, I'm not going to count anything which appears on your scratch paper for credit. Some of the following questions just show an answer, with no supporting work. That is not what you are to do on your own exam. For all open-ended questions, you need to show all the supporting work. Unsupported answers to open-ended questions on the actual exam are worthless.

68. The manager of a high-end gourmet coffee shop knows that if he sells a cup of hand-pour, a.k.a. pour-over coffee, for $4.50, the store will sell 1,600 cups per week. Some experimentation has shown that if they offer a buy-one-get-one-free coupon, the store will sell 2,400 cups per week. Assuming the number of cups sold (y) changes in a linear fashion according to the price per cup (x), answer the following questions.a) The buy-one-get-one-free coupon basically amounts to two cups for $4.50, which means the price per cup would be $2.25 with the coupon. Find the slope of the line containing the points (2.25, 2400) and (4.5, 1600).b) Find the equation of the line containing the points (2.25, 2400) and (4.5, 1600).c) Briefly explain the meaning of the slope, including the units.d) Briefly explain the meaning of the y-intercept, including the units.e) If they reduced the price per cup to $4.00, how many cups would they sell, according to the equation you found in part b)?f) If they sold 1,200 cups/week, what would the price be, according to the equation you found in part b)?


Consider the quadrilateral with vertices at A: (–2,3), B: (2,5), C: (4, 9), and D: (0, 7).

69. Is the quadrilateral given above a parallelogram? Justify your answer.

70. Is the quadrilateral ABCD a rhombus? Justify your answer.


71. Recall that a line tangent to a circle at a point is perpendicular to the radius drawn to that point (see the figure). Find the equation of the line tangent to the circle at the indicated point. Write the answer in the standard form Ax + By = C, A ≥ 0. Graph the circle and the tangent line on the same coordinate system. 2 2 225, 9,12x y

x

y

72. Write the equation in factored form to find the center and radius of the circle.x2 + y2 – 2x + 10y – 21 = 0

73. Determine whether the function 3 44 2f x x x

x is even, odd, or neither. Show all work.

For a video that shows you how to answer this question, scan the QR code shown below:

74. R varies directly as x2 and inversely as t when R = 15 when x = 5 and t = 15. Find R when x = 9 and t = 11.

75. An open box is formed from a rectangular piece of cardboard that measures 22 by 36 inches. Squares, x inches on a side, will be cut from each corner, and then the ends and sides will be folded up. Find a formula for the volume of the box V in terms of x. From practical considerations, what is the domain of the function V?


Sample Bonus Questions

On each exam, I like to include some optional questions that allow people with a superior mastery of the subject material to demonstrate this. I call these questions "bonus questions," and I handle them in the following way: if you attempt any bonus question and get it wrong, I won't take off any points. If you get it right-- it has to be 100% correct-- you can earn 10 bonus points per question, and you can submit up to two bonus problems per exam. Anything you submit in response to a bonus question needs to be "camera ready," meaning it needs to be clear, relatively free of confusion, and it needs to read like one of the solutions you find in the text-based examples and/or your solution manual. If you're unwilling or unable to write things up in this fashion, feel free to skip these questions.

76. Factor completely, first using the difference of squares pattern, then using the difference of cubes pattern: 6 1264x y

77. Factor completely: 2 216 40 25x x y .

78. Express the domain of the function 4 3f x x x using interval notation.

79. If the range of ( )h x is { : 4 3}y y find the range of 2 ( ) 1.h x

80. Suppose that f x is even and g x is odd. Determine whether the following functions are even, odd,

or neither. Justify your answer.A) f x g x

B) f x g x

C) f x g x

D) 1

f x g x

f x g x


Answer Key

1. e2. c3. b, c, d4. a, c, d5. a6. a, b7. d8. a9. c

10. c11. b12. d13. d14. a15. d16. e17. c18. a19. c20. b21. b22. b23. c24. b25. c26. b27. a, b, c, d28. d29. a30. d31. c32. e33. c34. e35. d36. a37. a38. d39. b40. a41. d42. b43. d44. e45. b46. d47. e48. a


49. a, b50. e51. a52. b53. e54. b55. e56. b57. d58. d59. b60. c61. d62. b63. e64. e65. b66. e67. d

68. a) The slope is 1600 2400 800 355.5

355.54.5 2.25 2.25 1

b) The equation is most easily expressed in the y mx b form. Picking one of the points-- it really doesn't matter which one, but I will choose (4.5, 1600)-- we have

1600 355.5 4.5 1600 1600 3200b b b , so the equation would be 355.5 3200y x ,

and if you're careless about using your calculator/rounding, etc., you'll probably have a difficult time getting that exactly. You'll be in the ballpark, but not on the money. The key here is to use the rational form of the slope in the calculation instead of a rounded decimal approximation.

c) The slope, found in part a), is 800 355.5

355.52.25 1

, and the units attached to it would be

units cups of coffee

unit dollar

y

x . Practically speaking, it means that for every $1 increase in the price of a cup,

on the average, 355.5 fewer cups will be sold.d) The y-intercept is 3200, or more specifically 0,3200 . This means that if the price of a cup was $0,

according to the model, 3,200 cups would be "sold" each week. In other words, if they gave the coffee away, people would consume 3,200 cups/week. Of course, this is an extrapolated value (outside of the data, predicted by the model, not necessarily corresponding to reality), but we can still give a meaningful explanation.

e) Since 355.5 3200y x , we have 355.5 4.0 3200 1777.7y , meaning they would sell

1777.7 cups, according to the model.

f) You are being asked to solve 1200 355.5 3200 2000 355.5 5.625x x x ; the price per cup, rounded to the nearest cent, would be $5.63.

69. The easiest way to show this is to calculate the midpoints of the diagonals. If they bisect each other (i.e., share the same midpoint), the vertices form a parallelogram. In this case, since the midpoint of

–2 4 3 9 1 6, ,

2 2 1 1AC

and the midpoint of 2 0 5 7 1 6

, ,2 2 1 1

BC

, we can see the figure

is a parallelogram.


70. A square is a special case of a rhombus, and in general, a rhombus is a parallelogram with four congruent sides. Algebraically, the best way to check if a parallelogram is a rhombus or not is to find whether the diagonals are perpendicular bisectors of each other. Assuming ABCD is a parallelogram,

find the slopes of the diagonals AC and BD . If the product of the slopes is 1 , the diagonals are perpendicular, and the figure is a rhombus. In this case, since

9 3 7 5 1 –1slope of AC slope of BD –1

4 –2 0 2 1 1

the figure is a rhombus.71. 3x + 4y = 75

-15

-10

-5

5

10

15

-15 -10 -5 5 10 15

x

y

72. (x – 1)2 + (y + 5)2 = 47; center (1, –5), radius 47

73. It helps to notice that 3 34 44 2 4 2f x x x x x

x x

. Then, since

3 34 44 2 4 2f x x x x x f x

x x

, it can be seen that f x is odd.

Even better, you can use the definitions of even functions and odd functions to show that sums/differences of even functions are again even, and likewise, sums/differences of odd functions are again odd. In this case we have a sum/difference of odd functions. For example, say you're given two odd functions, f x and g x . How can we show that the sum or difference of two odd functions is

again odd? First, recall that f x being odd means f x f x , and similarly with respect to

g x . So let h x f x g x . Then

h x f x g x f x g x f x g x h x ,

so that we have h x h x , meaning h x is odd. The argument is basically the same if we take

the difference of odd functions, i.e., if h x f x g x , or if we take the sum or difference of even

functions. Revision Date: Spring 2018

74.729

11R

75. V(x) = x(22 – 2x)(36 – 2x); Domain: 0 < x < 1176. Using difference of squares:

2 26 12 3 6 3 6 3 6

3 33 32 2

2 2 2 4 2 2 2 4

64 8 8 8

2 2

2 4 2 2 4 2

x y x y x y x y

x y x y

x y x xy y x y x xy y

Using difference of cubes:

3 36 12 2 4 2 4 4 2 4 8

2 2 4 2 4 8

64 4 4 16 4

2 2 16 4

x y x y x y x x y y

x y x y x x y y

Comparing the two answers, we can see that this is one instance where the quadratic that emerges from the difference of cubes pattern can actually be factored, since

2 2 4 2 2 4 4 2 4 84 2 4 2 16 4x xy y x xy y x x y y

77. Factoring by grouping doesn't always mean grouping the terms two-by-two.

22 2 2 2 216 40 25 16 40 25 4 5

4 5 4 5

x x y x x y x y

x y x y

78. 4, ,3 4,3 79. { : 7 7}y y

80. Recall, if f x is even, then f x f x , and if g x is odd, then g x g x .

A) Neither. Let h x f x g x ; then h x f x g x f x g x , which is neither

h x , which would mean h x is even, nor h x , which means it's not odd either. So it is neither

even nor odd.B) Neither. Let h x f x g x ; then h x f x g x f x g x , which is neither

h x , which would mean h x is even, nor h x , which means it's not odd either. So it is neither

even nor odd.C) Odd. Let h x f x g x ; then h x f x g x f x g x f x g x h x ,

which means it's odd.

D) Neither. Let 1

f x g xh x

f x g x

; then

1 1

f x g x f x g xh x

f x g x f x g x

, which is

neither h x , which would mean h x is even, nor h x , which means it's not odd either. The key is

to recognize (from part C above) that 1 f x g x is neither even nor odd, because translating the

function vertically causes it to lose origin symmetry ("oddness"). So this quotient is neither even nor odd. Convince yourself by writing down h x if necessary.


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