Summer Assignment for Functions/Analytic Geometry

Due Tuesday 9/4/18 or Wednesday 9/5/18 (depending on which day class meets)

This summer assignment is designed to prepare you for Functions / Analytic Geometry.

Nothing on the summer assignment is new. Everything is a review of topics students learned in

Algebra I & II/trig and Geometry. However, admittedly, there may be some problems involving

trigonometry that you may not be comfortable with. If you want to be successful during Functions /

Analytic Geometry, you must be able to understand and apply this information throughout next year.

The assignment may be completed with another student but be certain that YOU understand how to

complete every problem. Neatly show all work for each problem, using a pencil. There are less than 5

problems that require a calculator on this assignment (mainly in the trigonometry section) so do not

rely on it. During the first or second week of school, you will have an assessment covering all the

material from the summer assignment. We will review, but briefly and only on selected topics.

If you need to review these topics or see examples, I recommend the website

www.purplemath.com/modules/index.htm, which lists many Algebra review topics. The assignment

should be completed and brought to school on the first day of class and will count for extra credit on

your 1st marking period grade if completed correctly and on time:

+2 to 1st Quarter

Grade

Turned in on Due Date 100% complete with sufficient justification for

every problem.

+1 to 1st Quarter

Grade

Turned in on Due Date with more than 75%, but less than 100%

complete with sufficient justification, OR turned in one class late with

100% complete with sufficient justification for each problem.

+0 to 1st Quarter

Grade

No assignment turned in OR less than 75% problems completed OR

insufficient justification for several problems OR assignment turned in

more than 1 day late.

Simplify the following expressions using the properties above. Leave no negative exponents.

1. 14 −w

2. )5)(3( 82 ww−

3. 312 )2( −− zyx 4.

2004 )6()22( xx

Properties of Exponents

n

n

x

aax =−

mnnm xx =)(

nmnm xxx +=))(( 10 =x

PurpleMath Topics

Beginning Algebra Topics:

• Exponents:

o Basic Rules

o Negative Exponents

• Simplifying with Exponents

5. 11

5

64 p

a− 6.

52

7

80

64

ty

x−

−

Simplify the following expressions using the property above. Express radicals as fractional exponents.

7. 3 m 8. 5 4n

9. 5t

10. 65

15

−b

11. −3 √𝑘84 12.

7 √𝑚412

14

Simplify the following expressions using imaginary numbers.

13. 25−

14. 8−

15. 2i

16. 7√−72

17. 3

𝑖

18. 6

2+𝑖

Rational Exponents

n

m

n m xx =

Complex Numbers

i=−1

PurpleMath Topics

Beginning Algebra Topic:

• Exponents

Fractional Exponents

PurpleMath Topic

Advanced Algebra Topics

• Complex Numbers

Solve the following quadratic equations using the quadratic formula. Simplify as much as possible.

You will potentially have complex solutions.

19. yy 2112 2 =−

20. 0722 =++ cc

21. 6114 2 −=+ ww

22. 𝑝2 + 16 = 0

Quadratic Formula

a

acbbx

2

42 −−=

Factoring

Always look for a greatest common factor first:

)1(2 +=+ aababba

Perfect Square Trinomials: 222 )(2 bababa −=+−

or 222 )(2 bababa +=++

Difference of Squares: ))((22 bababa +−=−

Sum of Cubes: ))(( 2233 babababa +−+=+

Difference of Cubes: ))(( 2233 babababa ++−=−

PurpleMath Topic

Intermediate Algebra Topics:

• Quadratic Formula

PurpleMath Topics

Beginning Algebra Topics:

• Simple Factoring

Intermediate Algebra Topics:

• Factoring Quadratics

• Solving Quadratic Equations

• Special Factoring Formulas

Solve the following equations using factoring. Show factored equation and give all solutions (real and

imaginary).

23. 0714 2 =− xx 24. 092 =−x

25. 02163 =−x 26. 04129 2 =+− xx

27. 010293 3 =+x 28. 02564 =−x

29. 0352 23 =−− ppp 30. 031612 2 =−+ xx

31. 2090192 =+− pp 32. 0217824 2 =−− nn

Solve for the unknown in each of the figures below. Give answers accurate to three decimal places.

33.

34.

Use the following right triangles to find the value of the unknown. Give answers accurate to three

decimal places.

35.

36.

37. 38.

Pythagorean Theorem 222 cba =+

sin 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

cos 𝜃 =𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 tan 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

r°

68 cm

51 cm

PurpleMath Topic

...and the beginnings of trig:

• Pythagorean Theorem

PurpleMath Topic

...and the beginnings of trig:

• Basic Trigonometric Ratios

• Inverses of Trigonometric Ratios

• Special Angle Values

4 in.

3 in. x

4 ft 3 ft

b

15

m

x

33°

17 mm

x°

z

p 9 yd

26°

For each of the following sets of points, find:

a. the slope

b. the equation of the line that fits these points

c. the midpoint

39. )0,6()8,2(−

40. )8,10()4,5( −−−

41. )3,0()4,0(

Lines

slope: 12

12

xx

yym

−

−=

midpoint:

++

2,

2

2121 yyxx

equation of a line: bmxy += or )( 11 xxmyy −=−

PurpleMath Topics

Beginning Algebra Topics:

• Slope of a Straight Line

• Midpoint Formula

• Straight-line Equations

Expand the following expressions.

42. 2)8( x+ 43. 2)42( −x

Simplify the following rational functions. State any restrictions on the domain.

44. 34

62

2

+−

−−

xx

xx

45. 9

819

8118

6222

2

−

+

++

−

x

x

xx

xx

46. 11

3

−+

+ x

x

x

47.

yx

yx

yx

55

18

66

−

−

+

48. 1

11

−

−

x

x

Rational Expressions

0)(,0)(,0)(

,)(

)(

)(

)()(,

)(

)(

)(

)(

)(

==

xrxsxqwhere

xr

xs

xq

xpxfthen

xs

xr

xq

xp

xfIf

Binomial Expansion

222 2))(()( bababababa ++=++=+

PurpleMath Topic

Beginning Algebra Topics:

• Polynomials: Multiplying

PurpleMath Topics

Advanced Algebra Topics:

• Rational Expressions: Simplifying

• Rational Expressions: Adding

• Rational Expressions: Multiplying

• Complex Fractions

Library of Functions: These are all parent functions you should be familiar with for this course.

Graph each and complete the table below. Set notation and interval notation are two different ways of

presenting a set of numbers – research the difference if you don’t remember. 𝑓(𝑥) = [𝑥] represents

the step function, and “c” in #2 represents an arbitrary constant. Write the common name for each

function in the blank above the function.

Graph the following functions. Use at least 5 points, labelling each axis to scale. State the domain and

range of each.

49. 2)3(2)( −= xxf

50. xxf 2)( =

54. Solve the system using substitution:

4x + y = 9

3x – 2y = 4

55. Solve the system using elimination:

2m – n = -1

3m + 2n = 30

Graphing:

Changes to the “outside” of )(xf affect a graph vertically.

Changes to the “inside” of )(xf affect a graph horizontally.

Domain is the set of possible x-values.

Range is the set of possible y-values.

Systems of Equations

The solution to a system of equations,

)(

)(

xg

xf,

is the point of intersection, (x, y), of the functions.

PurpleMath Topics:

Intermediate Algebra Topics:

• Domain and Range

• Graphing Quadratic

Equations

PurpleMath Topics

Advanced Algebra Topics:

• Solving Systems of Linear Equations

Sections:

Substitution (p. 4)

Elimination/addition (p. 5)

Decide whether each of the following simplifications is accurate. If it is not, correct the right side of

the equation to make it true. If the left side cannot be further simplified, write “simplified. If you’re

stuck, try to substitute real numbers to see if the equation is true.

56. 532 9

?33

57. ( )?

3 34 4a a

58. -24?

16

59. 3x + 4y ?

7xy

60. 22 4

?)2( xx −−

61. 2 6 2 2 33 2 2x x x x x x− + −?

( )

62. )5)(5(?

)25( 2 +−+ xxx

63. 4?

)2( 22 ++ xx

64. 312?

432

65. xyyx?

−−

66. 2?

42 ++ xx

67. 52?

2

104−

−

68. x

x3

1?3 1−

69. xx ?

2

2 +

70. 1?

1

12

−+

−x

x

x

71. a b

x

a

x

b

x

++

?

72. a

x b

a

x

a

b++

?

73. 4

2?

20

105 ++ xx

74. 1?

21

12−

−

−

x

x 75.

x

x

x 3

2?

3

4

3

2 −−

Common Mistakes in Algebra

76. ay

ax

y

xa

?

77. 3

2)2(?

6

842 2

2

23

+

+−

−+

−+−

x

xx

xx

xxx

78. 2

1?

22

13

2

34

23

+

+

−+−

−+−

x

x

xxx

xxx

79.

+

+a

b

w

z

y

x

b

a

z

w

y

x

?

Trigonometry Review

80. Rewrite a degree measure of –1080° in

radians. State your answer in terms of 𝜋.

81. Convert the radian measure of 3𝜋

5 to degrees.

82. Sketch an angle of –150º in standard

position. This means that one ray of the

angle lies on the positive x-axis.

83. Determine the measure of the angle below

84. Find the measure of an angle between 0º and

360º coterminal with an angle of –110º in

standard position.

85. Find the exact value of sin (𝜋

3)

20

x

y

86. Find the amplitude and period of the sine curve shown below.

87. Find the exact values of and .

Graph the function in the interval from 0 to 2. Draw and scale your own axes.

88. 𝑦 = 2 sin 𝜃

89. 𝑦 = cos (1

2𝜃) − 1

Find the exact values for the following trigonometric expressions:

90. cos 𝜋 91. sin𝜋

6

92. sin2𝜋

3

93. cos𝜋

4 94. sin

𝜋

4 95. sin 2𝜋

96. Suppose tan 𝜃 =8

15. Find cot 𝜃 and sin 𝜃

2O

3

2

4

–2

–4

y

97. Find the exact value of csc 135º. If the

expression is undefined, write undefined.

98. Find the value of csc , if cos = ;

180 < < 270 .

99. Write an equation of the cosine function

with amplitude 2 and period 4.

100. Find the values of the six trigonometric functions for angle , when AC = 10 and BC = 8.

101. Use the Law of Cosines. Find to the nearest tenth of a degree.

102. Find the value of the trigonometric expression: cos 150° cos 120° – sin 150° sin 120°

B A

C

17

30

22