Valley High School

AB Calculus 2018 Summer Work

Attached you will find a variety of review work based on the prerequisites needed for the AB Calculus

curriculum. The problems assigned should be the minimum you do to be successful in your next course.

It is expected that you understand and can do all the attached. If you don’t understand any topic I advise

you to go on line to Khan Academy for help. I am also including some extension work for those of you

who would like to get a preview of the first chapter in the book.

Recommended Work: 1. Complete the Algebra Review

2. Complete the Trig Review (A-F)/ Trig Formula Sheet

Extension Work: 1. Introduction to Calculus: Limits, continuity, and derivatives.

(The topics on these sheets align to section 12.1-12.3 in your Pre-Cal Book)

Algebra Review

To help you prepare for the first semester of Calculus w

1. Quadratic Review 2. Polynomial Review

3. Logarithmic Review 4. Essential Skills Review

The Quadratic Review

Factor:

1. 2 9 20x x 2. 29 34 8x x 3. 263 175x 4. 24 52 25x x

State the discriminate and describe the nature of the roots:

5. 24 12 9 0x x 6. 2 3 9 0x x

7. Solve by completing the square. 2 6 12 0x x

Solve by Factoring

8. 25 7 6 0x x 9. 29 49 0x 10. 2 8 7 0x x

Solve using the quadratic formula.

11. 2 4 10 0x x 12. 2 6 3 0x x 13. 22 4 5x x

Place the following quadratic function into vertex form and state the vertex:

14. 2 8 1y x x 15. 23 12 4y x x

Graph the following parabolas

16. ( 4)( 2)y x x 17. 22 16 13y x x 18. 22 1y x

The Polynomial and Quadratic Review

Add/subtract,and or multiply the following

19 3 2 3(6 22 3 10) 4(3 2)x x x x x 20. 3 2 2( 5 13 7) 5(3 2)x x x x x

21. 2(2 3)(2 7 1)x x x

Divide the following

22. 3 2(6 22 3 10) (3 2)x x x x 23. 4 3(2 13 12 5) ( 5)x x x x

Solve. Find all answers, real and imaginary.

24. 316 2 0x 25. 3 0x x

26. 4 23 4 0x x 27. 4625 1 0x

28. 5 32 18 40 0x x x 29. 3 23 2 24 16 0x x x

Logarithmic Review

Simplify:

30. 1/532 31. 1/ 249 32. 1/327 33. 2/327 34.

1/31

8

35.

1/ 41

16

36. 3/ 416 37. 3/29 38. 2/3343 39. 2/532

Expand and/or Condense – simplify when necessary:

40. 2

3

3log

5

x

y

41. 3log2 log 2log7x 42. 3ln ln 4 3ln 3ln 2x y

Evaluate without a calculator:

43. 45lne 44. 2log 16 45. 27log 3 46.

3

1log

9

Solve. No calculator. Check for extraneous solutions.

47. 1

log 2100

x 48. 1

3

log 4x 49. 9

1log

81x 50.

2

2 1 12

4

x

x

51. 5 5log (5 3) log ( 11)x x 52.

4 4log (13 12) log ( 8) 2x x

Algebra 2 Essential Skills Review

***Drawings are not to scale. The graded mastery quiz will only include one of each type of question,

additional problems are provided to give a sense of the variations each question might contain.***

1. (Geo. St. 17) Calculate the midpoint of and distance between points A(4, 3) and B(7, -9).

2. (Alg. 2 St. 25) Evaluate the following for the given functions .

a. 3 4, find f a+4f x x b. 23 4 2, find f 5-w .f x x x

3. (Geo. St. 20) Use special right triangles to find the exact value of x and y for the given triangles.

a. b.

4. (Alg. St. 7&8) Find the equation of the line which passes through the point ( 3, -1) and is perpendicular to the line determined by

y x1

74

. Write your answer in both point-slope form and slope-intercept form.

5. (Alg. St. 9) Solve the following systems of equations algebraically.

a) 5 6 14

3 4 16

c d

c d

b)

3 5 12

2 9

x y

x y

6. (Alg. 2 St. 1) Write the Absolute Value Inequality that represents the given graphical solution.

a) b)

7. (Alg. St. 17) Answer the following for the given graphs:

a) Type of function/relation b) Domain. c) Range.

-7 3 3 6

1 2

3

30o

60o

x 18

y

30o

60o

45º

1

1 2

45º

x

y 12

8. (Alg. 2 St. 8) Find the roots of the following equations over the complex numbers.

a) 4 2 20 0x x b)

3 24 12 9 27 0x x x c) 6 0x x

9. (Alg. 2 St. 6) Simplify each expression

a) 5

3 7 b)

4 3

7 7

i

i

10. (Alg. 2 St. 24) Given 2 7 12 and 2, find and g f .f x x x g x x f g x x

11. (Alg. 2 St. 24) Find the inverse of each function listed below.

a) 3 7f x x b) 3 8g x x c) 2 2 8h x x x

12. (Alg. 2 St. 7) Simplify the following expressions, recall there are to be no negative exponents in final answer.

a) 2 3

2 4 32 2x y xy

b)

34 2

3 7

32

6

x y

y x

Problems 13-15 are Graphing Calculator Problems

13. (Alg. 2 St. 3) For the following find the zeros, max/min, the regions where f(x) is increasing and decreasing, and the regions

where f(x) is positive and negative

a) 4 26 8 3y x x x

14. (Alg. St. 2) Solve for x algebraically or graphically.

a) 3 6 2x x x b)

2

23

4 4

2

y x x

y x

15. (Alg. 2 St. 8) Complete the square and answer the following for the given quadratic.

a. Write in vertex form. b. max/min c. find zeros in exact form

a)22 16 33y xx b)

2 8 5y x x

Answers to Review problems The Quadratic Review

1. 4 5x x 2. 4 9 2x x 3. 7 3 5 3 5x x 4. 2 1 2 25x x

State the discriminant and describe the nature of the roots:

5. 0, 2 real rootsD 6. 0, 2 imaginary rootsD

7. Solve by completing the square. 3 3x i

Solve by Factoring 8. 3

, 25

x 9. 7

3x 10. 7,1x

Solve using the quadratic formula. 11. 2 6x i 12. 3 6x 13. 2 14

2x

Place the following quadratic function into vertex form and state the vertex:

14. 2

4 15, V:(4,-15)y x 15. 2

3 2 16, V:(-2,-16)y x

Graph the following parabolas

16. ( 4)( 2)y x x 17. 22 16 13y x x 18. 22 1y x

2

1 9y x 2

2 4 19y x

The Polynomial and Quadratic Review

Add/subtract,and or multiply the following

19 3 218 22 7 2x x x 20. 3 210 8 17x x x 21. 3 24 20 23 3x x x

Divide the following

22. 22 6 5x x 23. 3 2 28102 23 115 563

5x x x

x

Solve. Find all answers, real and imaginary.

24. 1 3 1

,4 2

ix

25. 0, 1x 26. , 2x i

27. 1 1

, 5 5

x i 28. 5, 2, 0x i 29. 2

2 2, 3

x

Logarithmic Review

30. 1

2 31. 7 32. 3 33.

1

9 34.

1

2 35. 2 36. 8

37. 27 38. 1

49 39.

1

4 40. log3 2log log5 3logx y 41.

8log

49x

42. 3

3

2ln

x

y

43. 20 44. 4 45. 1

6 46. - 4 47. 10 48.

1

81

49. – 2 50. 3

4

51.

7

2 52.

140

3

Algebra 2 Review SAMPLE MASTERY QUIZ

1. 153 11

, 32

2. a. 3 16a b. 23 34 93w w

3. a. 9 y=9 3x b. y=6 2x 4. y+1=4 x-3 , y=4x-13

5. a. 4,-1 b. 69 -15

,11 11

6. a. x+2 5 b.9

x+ 32

7.

i) a) Radical ii) a)Quadratic iii) a) Absolute Value

b) x 1, b) x , b) x ,

c) y 0, c) y ,8 c) y -6,

8. a) 5, 2x i b) 3

, -32

x c) 9x

9. a) 5 3 35

46

b)

1 7

14

i

10. 2 2= x -3x+2 and g f x -7x+14f g x x

11. a) 1 7

3

xf x

b) 1

1 38g x x c) 1 1 9h x x

12. a) 172

x

y b)

9

15

27

4096

x

x

Honors Precalculus Name__________________________________

Trigonometry Review: A to F Date __________________ Per. ____________

1. Sketch the following in standard position and find their complimentary and supplementary angles:

a. 87 b. 2

5

c. 128

2. Sketch the following in standard position and find a positive and negative co-terminal angle:

a.13

9

b. 234 c.

5

8

3. Sketch the following in standard position and find their reference angles:

a.3

10

b. 333 c. 222 d.

11

12

For each of the given trigonometric functions: draw the angle, list the reference angle, and find the desired ratio.

4. sin 240 ____ ref. angle:____ 5. cos150 ____ ref. angle:____

6. tan 225 ____ ref. angle:____ 7. tan 600 ____ ref. angle:____

8. cos315 ____ ref. angle:____ 9. sin 405 ____ ref. angle:____

0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180°

0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180°

0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180°

10. csc330 ____ ref. angle:____ 11. cot 240 ____ ref. angle:____

12. sec( 120) ____ ref. angle:____ 13. cot( 225) ____ ref. angle:____

14. 11

sin6

____ ref. angle:____ 15.

5cos

4

____ ref. angle:____

16. 4

tan3

____ ref. angle:____ 17.

7tan

4

____ ref. angle:____

18. 3

cot4

____ ref. angle:____ 19.

5sec

3

____ ref. angle:____

0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180°

0°(360°)

90°

270°

180° 0°(360°)

90°

270°

180°

20. sin( )6

____ ref. angle:____ 21.

5cos

4

____ ref. angle:____

22. 17

tan6

____ ref. angle:____ 23.

15csc

4

____ ref. angle:____

Given the following information, determine the quadrantal angle in both radians and degrees.

24. cos 0 and sin 1

______ or _______

25. tan 0 and cos 1

______ or _______

26. tan is undefined, and sin 1

______ or _______

27. sin 0 and cos 0

______ or _______

28. cos 0 and csc 1

______ or _______

29. tan 0 and sec 1

______ or _______

Find the values of the six trigonometric function of .

30. 4

cos5

in QIII 31. 15

tan8

and sin 0

32. csc 4 and cot 0 33. 5

sec3

and csc 0

34. sin x in QI

For each of the following trigonometric ratios and quadrants determine the reference angle (in radians) that would give that

ratio and the actual value of (in radians).

35.3

sin2

in QII. 36. 2

cos2

in QI

ref. angle:_______ ______ ref. angle:_______ ______

37.1

tan3

in QIII. 38. 3

cos2

in QII

ref. angle:_______ ______ ref. angle:_______ ______

39. tan 3 in QIV. 40. 3

cot3

in QIV.

ref. angle:_______ ______ ref. angle:_______ ______

41. 1

cos2

in QIII 42. csc 2 in QI

ref. angle:_______ ______ ref. angle:_______ ______

For each of the following trigonometric ratios and quadrants determine the reference angle (in degrees) that would give that

ratio and the actual value of (in degrees).

43. 2

cos2

in QIII 44. 3

cos2

in QI

ref. angle:_______ ______ ref. angle:_______ ______

45. tan 3 in QIII. 46. 1

cos2

in QII

ref. angle:_______ ______ ref. angle:_______ ______

47. tan 1 in QIV. 48. 1

sin2

in QIV

ref. angle:_______ ______ ref. angle:_______ ______

49. csc 2 in QI. 50. 3

cot3

in QIV.

ref. angle:_______ ______ ref. angle:_______ ______

Honors Pre-Calculus Name_____________________________________

Trig. Identity Sheet Date______________________ Per. ___________

Reciprocal Identities: The following Trig identities are true for all values of , except those for which the function is

undefined:

sin csc =

cos sec =

tan cot =

Quotient Identities: The following Trig identities are true for all values of , except those for which the function is

undefined:

tan cot =

Co-function Identities: The following Trig identities are true for all values of , except those for which the function is

undefined:

sin and tan and sec2 2 2

cos and cot2 2

and csc

2

Even and Odd Identities: The following Trig identities are true for all values of , except those for which the function is

undefined:

sin( ) cos( ) tan( )

Pythagorean Identities: The following Trig identities are true for all values of , except those for which the function is

undefined:

1. or 2sin or

2cos

2. or 2tan

3. or 2cot

Sum & Difference Identities:

cos cos

sin sin

tan tan

The Double Angle Identities:

sin2 tan2

cos2 or cos2 or cos2

Half Angle Identities:

cos2

sin

2

tan2

or tan

2

Power Reducing Identities:

2cos 2sin 2tan

Product-to-Sum Identities:

1

sin sin cos cos2

u v u v u v 1

cos cos cos cos2

u v u v u v

1

sin cos sin sin2

u v u v u v 1

cos sin sin sin2

u v u v u v

Sum-to-Product Identities:

sin sin 2sin cos2 2

x y x yx y

sin sin 2cos sin

2 2

x y x yx y

cos cos 2cos cos2 2

x y x yx y

cos cos 2sin sin

2 2

x y x yx y