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Name: Rising Math Analysis Summer packet for 2017-18

These are all things students should know coming into Math Analysis. Math Analysis is a fast-paced course with many units. It is designed to prepare students for the AP Calculus BC course. It is important that students maintain all pre-requisite skills. Students should make sure they are comfortable with this material and be prepared for a quiz on the unit circle the first week of school.

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Directions: Please answer the following questions and show your work. Write your answer in the blanks. If an angle is given in radians, answer should be in radians. If an angle is given in degrees, answer should be in degrees. Leave all answers as exact, meaning in radical form, no decimals, unless otherwise specified. Be sure to simplify radicals. Students should answer all questions without a calculator.

1. Determine the reference angle, θ, for the following angle measures of θ from standard position. (Note: You are NOT turning degrees to radians and vice versa. Recall what a reference angle is.)

285°

θ′ = ________

8

7

θ′ = ________

3

4

θ′ = ________

320°

θ′ = ________

2. Determine the exact values of the six trigonometric functions of an angle of 330°. cos θ = _____

sin θ = _____

tan θ = _____

sec θ = _____

csc θ = _____

cot θ = _____

3. In which quadrant does the terminal side of θ lie from standard position? tan θ < 0 and sin θ > 0

0 < θ < π and cot > 0 3sec and 2

6csc

72sin and sec θ < 0

54tan and 22

cos θ > 0 and csc θ < 0

4. Find the values for x, y, and r for each of the following situations.

117co s & θ in quadrant

II

x = ____

y = ____

r = ____

85sin and 0 < θ <

2

x = ____

y = ____

r = ____

49tan and sin θ > 0

x = ____

y = ____

r = ____

712csc and cos θ > 0

x = ____

y = ____

r = ____

65co t and csc θ < 0

x = ____

y = ____

r = ____

715sec and - 2

< θ < 0

x = ____

y = ____

r = ____

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5. Given the following information, find the exact value of the six trig functions. (-3, -3) is a point on the terminal side of θ.

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

( 53 , 2) is a point on the terminal side of θ.

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

The terminal side of θ intersects the unit circle

at

7

102,

7

3.

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

sin θ = 9

5 , and cos θ =

9

142.

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

2

3cos and tan θ < 0

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

3

7

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

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6. Graph the six basic trig functions and define their domain and range.

Graph Domain Range

f(x) = sin x

f(x) = cos x

f(x) = tan x

f(x) = csc x

f(x) = sec x

f(x) = cot x

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Directions: Solve each problem.

7. Find all the zeros of the function: 241632)( 34 xxxxf

8. Find all the zeros of the function: 310114)( 23 xxxxf

9. Divide by long division: 23

824 2

x

xx.

10. Divide by synthetic division: 2

8122 235

x

xxxx.

11. Use synthetic substitution to find )3(f if 1542016)( 23 xxxxf .

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12. Given the polynomial function: 72)( 24 xxxf . Find the complex zeros of the

function and write it in completely factored form.

13. Find the complex zeros of the polynomial function: 3 23 25 75g x x x x

14. Write the polynomial function that has the given zeros. 4, 3, 0 2, 4 + 𝑖

15. Evaluate each expression. Leave your answer as a fraction in reduced radical form.

𝑡𝑎𝑛 (−330°) sec ( −30°) cot

6

7 csc

3

2

𝑠𝑖𝑛−1(−1)

sin ( 𝑐𝑜𝑠−10)

2

2sinco s 1

2

3co scsc 1

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16. Use any of the angle formulas to evaluate the exact angle.

cos(165°)

12

11tan

𝑐𝑜𝑠(𝑡𝑎𝑛−1(−√3 ))

17. Find the exact value of the six trig functions.

csc 7 2

2 and cos > 0

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

The terminal side of intersects the unit circle

at

2 5

5,

5

5

sin θ = ______ cos θ = ______ tan θ = ______ csc θ = ______ sec θ = ______ cot θ = ______

18. True or False – Explain why.

a) 3

2

3

2coscos1

b) 3

2

3

2si ns i n1

c) 2

1

2

1coscos 1

d)

tan1 tan2

3

2

3

8

x

y

x

y

x

y

x

y

19. State the amplitude, period, and phase shift for each function.

1 1 5

csc6 3 8

y x

Amplitude: Period: Phase Shift:

𝑦 = 3 sin (1

2𝑥 − 𝜋) + 2

Amplitude: Period: Phase Shift:

𝑓(𝑥) = cos (3𝑥 +𝜋

2)

Amplitude: Period: Phase Shift:

20. Graph each function and provide the domain and range in interval notation.

𝑓(𝑥) = 3 𝑐𝑜𝑡(1

2𝑥)

Domain: Range:

𝑔(𝑥) = 3 𝑠𝑒𝑐 (2𝑥)

: Domain: Range:

𝑓(𝑥) = 1

2 𝑡𝑎𝑛 (𝑥 +

𝜋

4) – 1

Domain: Range:

𝑔(𝑥) = 3 𝑠𝑖𝑛 (1

2𝑥 −

𝜋

2 )

Domain: Range:

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21. Find all solutions in the interval [0,2 ) 2sin2(x) – 5 sin(x) + 3 = 0

22. Find all the solutions in the interval [0,2 ) 2sin(x)cos(x) + cos(x) = 0

23. Verify the identity: sin ( – x) = sin (x)

24. Verify the identity:

)sin()tan(

)cos()sec(x

x

xx

25. Find the domain of 4217

9)(

2

xx

xxf . 26. Find the intercepts of

2

43)(

x

xxg .

x-intercept(s):_____________ y-intercept:________________

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27. Analyze the function, and sketch the

graph of 158

103)(

2

2

xx

xxxQ .

Domain:__________

Intercepts:_________

Symmetry:_________

Asymptote(s):_______

Intervals above x-axis:_______

Intervals below x-axis:_______

28. Analyze the function, and sketch the graph of

4

12)(

2

2

x

xxxR .

Domain:__________

Intercepts:_________

Symmetry:_________

Asymptote(s):_______

Intervals above x-axis:_______

Intervals below x-axis:_______

Exponential & Logarithmic Functions Review Guide

29. Directions: Describe as exponential growth or decay.

2( ) 5

3

x

f x

1 6( )

3 5

x

f x

5( ) 2

3

x

f x

21

( ) 32

x

f x

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30. Directions: Graph each function and identify its key characteristics. 1( ) 3 6xf x

Domain: __________________ Range: ___________________ Asymptote: _______________

Growth / Decay

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( ) 22

x

f x

Domain: __________________ Range: ___________________ Asymptote: _______________

Growth / Decay

2( ) log 3f x x

Domain: __________________ Range: ___________________ Asymptote: _______________

Growth / Decay

1

3

( ) log ( 2) 1f x x

Domain: __________________ Range: ___________________ Asymptote: _______________

Growth / Decay

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31. Directions: Rewrite the following. Write in logarithmic form.

28 64 42 32x 210 54x 2 15

25

Write in exponential form.

3log 27 3 1log 7

2x 4log 90 x ln 38x

32. Directions: Evaluate each logarithm.

9log 81 81log 3 5

1log

25 6log 1

33. Directions: Use the properties of logarithms. Condense each expression into a single logarithm.

3log2 log( 4)x

5 5

1log log 2

2x

33ln( 2) ln 4

2x

Expand each expression. 2 5 3

3log ( )x y

3

2ln

a

3 10

4log p q

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34. Directions: Solve the following equations.

4 4log (5 7) log (2 31)x x 7 5 364 4x x

1 65ae 2ln( ) ln(6 18)p p p

8 8 8

1log 36 log (3 7) log 132

2k

2

8 19

27

w

w

14

58 48n 2log( 5) log20 log5y

2 53 7 34me 42(3) 11 61y

2log (9 2) 7m 5ln(2 1) 15a

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35. Directions. Graph the following functions and provide the information. Write “none” on the line if there is not a VA, HA, or SA, etc.

2

2

2 15( )

36

x xf x

x

V.A.____________

H.A.____________

S.A._____________

Domain__________

X – int____________

Y- int_____________ Hole______________

2

3( )

2 8

xg x

x x

V.A.____________

H.A.____________

S.A._____________

Domain__________

X – int____________

Y- int_____________ Hole______________

16

22 4 6

( )3

x xf x

x

V.A. ____________

H.A.____________

S.A._____________

Domain__________

X – int____________

Y- int_____________

Hole _________________ 36. Simplify the following.

2

2

5 14

4 4

x x

x x

2

2

3 13 4 4 16

4 2

x x x

x x

2

2

2 2

4 5 5 25

x x x

x x x

2

2

5 2 32

4 16 25

x x

x x

17

9 2

3 1

x

x x

4 12

5 5 20

x

x x

16

24 6

1

x

x x

63

410

x

x

37. Solve the following. Check for extraneous solutions!

4 5

3 3x x

8 31

5x x

18

2

2

6 8 4

3 9 3

x x

x x x

3 4 1

2 1 1

x

x x

2

2 1

3 2 3x x x

2 1 1

3 3

x

x x x

2

5 32

6 2

x

x x x

2 2 3 5

1 1 1

x x

x x x

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Graph the following functions. List all information about asymptotes, intercepts, domain, and end behavior.

To find vertical asymptote – set denominator equal to 0

To find horizontal asymptote – look at the type of rational equation

To find x-intercepts – set the whole equation equal to 0 and solve for x. (x, 0)

To find y-intercept – substitute 0 in for x and solve. (0, y)

38. 3

(x) 14

fx

x-intercept(s) : _____________________

y-intercept: ________________

Vertical Asymptote(s): _______________________

Horizontal Asymptote(s): ________________

Domain: _______________________

End Behavior: as ,x f x _________

as ,x f x _________

39. 4

(x) 32

gx

x-intercept(s) : _____________________

y-intercept: ________________

Vertical Asymptote(s): _______________________

Horizontal Asymptote(s): ________________

Domain: _______________________

End Behavior: as ,x f x _________

as ,x f x _________

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40. 8 3

(x)2 6

xh

x

x-intercept(s) : _____________________

y-intercept: ________________

Vertical Asymptote(s): _______________________

Horizontal Asymptote(s): ________________

Domain: _______________________

End Behavior: as ,x f x _________

as ,x f x _________

41.

2

2

3 13 10(x)

5 4

x xk

x x

x-intercept(s) : _____________________

y-intercept: ________________

Vertical Asymptote(s): _______________________

Horizontal Asymptote(s): ________________

Domain: _______________________

End Behavior: as ,x f x _________

as ,x f x _________