June 2017 Dear Student: Welcome to Pre-Calculus. We commend you on your decision to enroll in this honors level class. The purpose of this course is to get you prepared for Advanced Placement Calculus. To help you prepare for the upcoming year and to ensure adequate time to cover the required curriculum, your first assignment is to complete the attached material. This assignment emphasizes important information from previous courses you will be need. We recommend you do not wait until the last part of summer to complete this assignment, but rather work on it regularly throughout the vacation. The assignment must be completed and submitted on the first day of school. It will be graded based upon completeness and accuracy of solutions as well as work shown. Points will be deducted for missing work. A penalty of five points per day will be imposed for every day the assignment is past due. Moreover, you will be tested on this information within the first few days of school. What to submit on the first day of school (all solutions must show detailed work): 1. Mixture of algebra problems: complete all problems with all work shown on separate sheets.

2. Composite functions: Study Guide and Assessment 1: check bottom of each page for the

problems you need to do starting on p.57.

3. Systems of equations and matrices: Study Guide and Assessment 2: check bottom of each page for the problems you need to do starting on p.119.

Resources: Your Algebra 2 notebook is a good resource if you get stuck on any problem. There are websites available for assistance as well. Many students find the websites www.purplemath.com helpful for researching and reviewing topics. Other excellent websites which include videos and step-by-step instructions on a wide range of topics are www.Kahnacademy.org and ck-12.org. We look forward to working with you in the upcoming school year. We hope you are prepared for a rewarding, yet challenging year. Enjoy the summer! Sincerely, Lewis Mills Mathematics Department

Note: this packet is also available online at: www.region10ct.org/LSM

Use separate sheets to show work. Full credit will only be given if work is shown.

1. Find the point-slope form of the equation of the line passing through the pair of points.

−6,−2 𝑎𝑛𝑑(5,3) 2. Why is it impossible for 𝑐𝑜𝑠 ∝= 2

3 but it’s okay for 𝑡𝑎𝑛 ∝= 2

3?

3. For the situation, find a linear model and use it to make a prediction.

The data below show hours spent researching the stock market per week and the percent gain for an investor.

What percent gain would an investor see after researching 21 hours per week? 4. No problem here, move on to the next one. 5. A rental car agency charges $20 per day plus 11 cents per mile to rent a certain car. Another

agency charges $24 per day plus 6 cents per mile to rent the same car. a. Write a system of equations to represent the cost c for each agency for m miles that

you need to pay daily. b. Find the number of miles for which the costs are the same. Round your answer to the

nearest tenth of a mile. 6. Identify the graph that shows the solution to the system of inequalities. Explain your answer

[A] [B]

[C] [D]

Hours 6 8 10 12 14 16 18%Gain 25 31.5 40.5 46 52.5 60.5 67

y xy x£ +³ - +RST

72 3

x

y

–10 10

–10

10

x

y

–10 10

–10

10

x

y

–10 10

–10

10

x

y

–10 10

–10

10

7. Pioneer Math Club must sell at least 8 school jackets and at least 21 caps during a fund-raiser. The club will make $16 profit on every jacket sold and $3 profit on every cap sold.

a. Write a system of inequalities that shows how many jackets and caps the club members need to sell to make a profit of at least $320.

b. Graph the system of inequalities from part (a). c. Will the club meet its goal of $320 profit if it sells 13 jackets and 36 caps? How much

above or below its goal will the club be? 8. Use the information in the table below.

SOURCE: The 1996 World Almanac

a. Put the information into two matrices. Label each matrix. b. b. Subtract the air distances to Lima from the air distances to Tokyo for each city

listed on the left. c. c. What does a negative value represent?

9. Skip this one 10. Skip this one too 11. Skip this one as well (I wonder why the numbers are used if there are no problems…hmmm) 12. Evaluate (find) the determinant of the matrix without using a calculator. For a 2×2 matrix,

the determinant is found by subtracting the products of its diagonals, expressed as: ad - cd

13. Suppose you want to make 15 pounds of snack mix for a meeting. You plan to buy dried

apricots for $.90/lb, dried blueberries for $1.20/lb, and peanuts for $1.70/lb. You have $21 and want the mix to contain as much of the nuts as of the fruit by weight. How much of each ingredient should you buy?

a. Explain how each equation in the system relates to the problem. What does each variable represent?

b. Solve the system. 14. Another mysterious missing question.

Air Distances Between Selected Cities miles

BangkokChicagoMexico CityMontrealStockholm

Tokyo Lima2868 12,2446314 37757035 26396471 39705053 7166

b g

5 1´

– –– –4 97 1LNM

OQP

x y zx y z

x y z

+ + =+ + =

+ =

RS|T|

150 90 12 17 21. . .

15. Solve the system using any method:

16. Solve the system using an augmented matrix or by using an inverse matrix. Show all work.

17. Find the quadratic model for this data:

[A] [B]

[C] [D] 18. Nothing to see here, move to the next problem. 19. In an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and

then warmed again.

a. Find a quadratic model for the data. b. Use the model to estimate the population of bacteria in 10 hours.

20. If two birds collide in the air, umm, never mind.. skip this one. 21. A manufacturer determines that the number of cabinets it can sell is given by the formula

where p is the price of the cabinets in dollars. a. At what price will the manufacturer sell the maximum number of cabinets? b. What is the maximum number of cabinets that can be sold?

22. Solve by factoring: 23. = 0. Show all work.

[A]

[B] 4, –6 [C]

[D] 2, –3

24. Solve the equation by finding square roots.

- - =+ =

RST8 4 52 8 0x yx y

–

x y zx y zx y z

+ + =+ + =- + =

RS|T|

2 93 7 4 28

2 2

––

xf x

– – – –4 3 2 188 50 22 4b g

f x x xb g = - -5 3 42 f x x xb g = - - +3 4 52

f x x xb g = - -5 4 32 f x x xb g = - + -4 5 32

Time (hr) 0 1 2 3 4 5 6Population (1000's) 5.8 3.69 2.16 1.21 0.84 1.05 1.84

D p p= + -– ,3 252 1602

x x2 4- = –3

3 22x x+ -

-23

1, 23

1, –

6 662x =

25. This table shows the height in feet of some of the tallest buildings in the United States.

The length of time it would take an object to fall from the top of one of these structures is where H is the height in feet of the structure, t is the number of seconds,

and h(t) is the height after t seconds. Find the time it would take an object to fall to the ground from the top of the John Hancock Center. Round your answer to the nearest hundredth of a second.

26. Simplify the number by using the imaginary number i.

[A] [B] [C] [D] 27. Simplify the expression.

4(1+7i)(8-5i) 28. Solve the equation. Be sure to show all work.

[A]

[B]

[C] [D]

29. Skipper-roo 30. Convertthefollowingintoexactvaluedegreesorradians.

Degrees 160° 80° Radians 3𝜋

4 7𝜋9

9𝜋5

31. Why so many skipped problems? I wonder what they could have been? 32. The daily profit of a custom T-shirt shop can be modeled by where

n is the number of T-shirts produced each day and is the profit made on that number. Rewrite this function in vertex form and determine the maximum daily profit.

33. Findthecenter,radius,circumferenceandareaofthecirclex2+y2–8x+4y=-4by

convertingtostandardform(hint:convertingtostandardformrequirescompletingthesquareforx-termsandy-terms).

Building City Height (ft)Sears Tower Chicago 1454Empire State Building New York City 1250AMOCO Chicago 1136John Hancock Center Chicago 1127Chrysler New York City 1046First Interstate World Center Los Angeles 1017

h t t H( ) = - +16 2 ,

-448

- 448i -8 7i 8 7i i -448

9 6 1 642x x+ + =

-653

21, - 3 73

, -11 5, 73

3,

P n n nb g = - + -2 60 400,P nb g

34. The motion of a ball scooped by a field hockey player can be modeled bywhere t is the time in seconds and h is the height of the ball. Will the ball ever reach 22 feet? If so, how many seconds will it take?

35. Divide using synthetic division. (You remember synthetic division, right?)

36. Factor the expression (hint: use substitution to made it a quadratic and then replace). 37-38 Simplify the rational expression. State any restrictions on the variable. Many of these can be factored which will result in terms that can be simplified. Show all work.

37.

38. . Be sure to show work

[A] ; [B] ;

[C] ; [D] ;

39. Multiply. State any restrictions (extraneous solutions) on the variables. Show all work

×

[A] ; x 5, 4, 0 [B] ; x –5, –4

[C] ; x 5, 4, 0 [D] none of these

40. Divide and simplify. State any restrictions on the variables. Show work.

41. Identify the largest angle of . How do you know

it’s the largest angle?

h t t= - +16 402 ,

x x x x x4 3 212 27 68 84 2+ + - - ÷ -c h b g

x x4 28 15- +

x xx

2 5 248- --

x xx x

2

2

10 217 12

- +- +

xx-+74x x¹ ¹–4, 3

xx--74

x x¹ ¹4 3,

xx++74x x¹ ¹–4 –3,

xx+-74x x¹ ¹4, –3

xx

2

5-x xx x

2

2

9 204

- +-

x ¹- +9 2020xx

¹

x xx

2 44

+-

¹

xx

xx

+-

÷--

11

11

2

DABC

A B

C

6 7

8

42. Combine fractions and simplify

43. Combine and simplify . Show all work.

[A]

[B]

[C]

[D]

Simplify the complex fraction to lowest terms showing all work.

44. [A]

[B] [C]

[D]

45. Solve the equation and check your answer.

46. Solve for x and check solution. =

47. Solve for x and check solution.

[A]

[B]

[C] [D] no solution

48. Another one of those mystery skips

49. This one too? Yup, it’s a skipper. 50. How long (years, months, days) will it take an investment of $15,000 to grow to be worth $35,000 if it

is invested at a rate of 4.25%, compounded continuously? Show work. (4 points)

51. Write 78Log in exponential form 52. Solve for x: 53. Solve for t: 54. Solve for x:

3 3 3 3log 5 log 5 log 1 log 3x+ = -

Something to ponder….what happened to all the skipped problems? Maybe they were not really there at all and were imaginary.

x xx x x

2

2

4 56 5

15

- -+ +

- +

xx

xx

x+-

++

24

2 39

5242

4 312 3

xx+ 15 16 6 36

72

3 2

2

x x xx

- - + 4 312 3

xx- 15 16 42 36

72

3 2

2

x x xx

- + +

1 35

3 17

+

+

750

6160 5528

2855

--

- +

41

3 5

x

x

xx-+34

xx--28

- + =2 45

1x x

–

56

65

-1

Onto Chapters 1 and 2 of your textbook. These topics include composite functions and matrices. You’ll need to know these thoroughly for what’s coming up in Pre-Calculus. Be sure to look at the bottom of each page for the problems that need to be completed. Not all problems are required (thank goodness!).

Do problem 1-10 on this page

Do problems 16, 22, 26, and 30 on this page.

Do problems 48, 50, 52 on this page

Skip this page

Do problems 67, 68, 69, and 2 on this page

Do problems 1-10 on this page.

Do problems 16, 18, 20, 22, 24, 26 on this page.

Do problems 34, 38, 40 on this page

Answers to Algebra Review portion of packet: 1.

2. cosine is defined as adjacent side/hypotenuse. The adjacent side cannot be longer than the

hypotenuse. Tangent is defined as opposite side/adjacent side. One leg can be longer than the other.

3.

4. skip

5. 80 miles

6. C

7.

8.

9. Skip 10. Skip 11. Skip 12. -59 13.

14. skip 15. 𝑥 = ?

3; 𝑦 = − ?

B2

16. (-5, -3, 2) 17. A 18. Skip 19. 𝑃 = 0.29𝑥B − 2.40𝑥 + 5.80 10.8 or 10,800 bacteria 20. 𝑠𝑘𝑖𝑝 21. a. $42 B. 5132 cabinets 22. x=3 and x=1 23. C 24. ± 11 ≈ ±3.32 25. 8.39𝑠𝑒𝑐𝑜𝑛𝑑𝑠 26. C 27. 172 + 240𝑖 28. B 29. Skip 30. 160° = 2L

M𝑟𝑎𝑑𝑠; 135° = OL

P𝑟𝑎𝑑𝑠; 80° = PL

M𝑟𝑎𝑑𝑠; 140° = 3L

M𝑟𝑎𝑑𝑠; 324° = ML

?𝑟𝑎𝑑𝑠

31. skip 32. 𝑃 𝑛 = −(𝑛 − 30)B + 500; $500𝑖𝑠𝑡ℎ𝑒𝑚𝑎𝑥𝑖𝑚𝑖𝑚𝑢𝑚𝑝𝑟𝑜𝑓𝑖𝑡 33. (𝑥 − 4)B + (𝑦 + 2)B = 16; 𝐶𝑒𝑛𝑡𝑒𝑟 4, −2 𝑟 = 4, 𝐶𝑖𝑟𝑢𝑐𝑚 = 8𝜋𝑢𝑛𝑖𝑡𝑠𝐴𝑟𝑒𝑎 =

16𝜋𝑠𝑞𝑢𝑛𝑖𝑡𝑠 34. yes; about 0.8 and about 1.7 seconds 35. 𝑥O + 14𝑥B + 55𝑥 + 42 36. C 37. – 𝑥 − 3; 𝑥 ≠ 8 38. 𝐵 39. A 40.

41. Angle C is largest because it’s across from the longest side. 42. \]^

\_?; 𝑥 ≠ −1𝑜𝑟 − 5

43. 𝐵 44. D 45. ]P\

?\`]2\_O

46. 𝑥 = OBaO

47. B 48. Skip 49. Skip 50. t= 19.936 years or 19 yrs 11 months 86.52 days 51. 7\ = 8 52. x=8 53. t is about 147.2 54. x = -7/5