TCS Math Summer Assignment 2013

The attached Math packet is assigned to you based on your registration for Pre-Calculus in the 2013-2014 school year.

Your assignment is to complete all of the problems in the packet (showing your work) and bring the completed packet to your Math class on the first day of school, Wednesday, September 4th. The packet will be checked for completeness at this time and you will receive a homework grade equivalent to three regular homework assignments.

Over the first two days of classes, you will have an opportunity to ask questions and we will review the assigned summer work. During the following week, you will be given a test on this material.

SUMMER MATH WORK FOR STUDENTS

ENTERING PRECALCULUS

SOME FORMULAS and FACTS NEEDED FOR DOING THIS MATH.

Factoring the Difference of cubes: a3 – c3 = (a – c) (a2 + ac + c2)Factoring the Sum of Cubes: a3 + c3 = (a + c) (a2 – ac + c2)

Arithmetic Sequences: un = u1 + (n-1)d Geometric Sequences: un = u1· r(n-1)

Arithmetic Series: Sn = n(u1 + un) Geometric Series: Sn = u1( 1 – r n )

2 (1 – r)Infinite Geometric Series: S∞ = u1

1 - rSlopes of lines: m = y2 – y1

x2 – x1

Equations of nonvertical lines: Equations of lines:Slope-intercept form: y = mx + b Standard form: Ax + By = C, wherePoint – slope form: y – y1 = m(x – x1) A, B, C are integers, A≥0, A and B

are not both 0.Two nonvertical lines are parallel if and only if they have the same slope. Two nonvertical lines are perpendicular if and only if the product of their slopes is –1.A vertical line has undefined slope and an equation of the form x = c, where c is a constant. A horizontal line has slope 0 and an equation of the form y = b, where b is a constant. Standard form of a quadratic equation: ax2 + bx + c = 0 For the form x2 + bx + c = 0, the sum of roots = –b , the product of roots = c . a a a a

_______Quadratic Formula: x = –b ± √ b 2 – 4ac 2a

Laws of Logarithms:Product Law Log b (MN) = Log b M + Log b NQuotient Law Log b (M/N) = Log b M – Log b NPower Law Log b Nk = k Log b NChange of Base Law Log b N = (Log d N)/ (Log d b)

Compound Interest Formulas where P = principal, r = annual interest rate in decimal form, and A = amount in account at the end of t years.

A = P(1 + r/n)nt for interest compounded n times per yearA = Pert for interest compounded continuously, e is Euler’s constant.

An odd function f has symmetry with respect to the origin (0,0) and f(–x) = –f(x).

An even function f has symmetry with respect to the y-axis and f(–x) = f(x). ___

The standard form of a complex number is a + bi, where a and b are real numbers and i = √ – 1.

Right triangle trig: In ΔABC where C = 90°, sin A = a/c, cos A = b/c, tan A = a/b (SOHCAHTOA)

To change degrees to radian measure, multiply the number of degrees by Π/180.To change radian measure to degrees, multiply the number in radian measure by 180/ Π.

Arc length L = rӨ, where r = the radius of the circle and Ө = the radian measure of the central angle.

Basic Trig Identities:

Reciprocal identities: sec Ө = 1/cos Ө, csc Ө = 1/sin Ө, cot Ө = 1/tan Ө

Ratio Identities: tan Ө = sin Ө , cot Ө = cos Ө cos Ө sin Ө

Pythagorian Identities: sin2 Ө + cos2 Ө = 1 tan2 Ө + 1 = sec2 Ө 1 + cot2 Ө = csc2 Ө

Negative Angle Identities: sin (– Ө ) = – sin Өcos (– Ө ) = cosӨ

tan (– Ө ) = – tan Ө

Law of Cosines for ΔABC: a2 = b2 + c2 –2bc cosA b2 = a2 + c2 –2ac cosB

c2 = a2 + b2 –2ab cosC

Law of Sines for ΔABC: = =

______________Area of a triangle ΔABC: √ s(s–a)(s–b)(s–c) , where s = (a+b+c)/2

Area of ΔABC = .5absinCArea of ΔABC = .5acsinBArea of ΔABC = .5bcsinA

FACTORING, BASIC OPERATIONS, SOLVING EQUATIONS AND INEQUALITIES

# 1 – 9: Factor completely over the set of rational numbers.

1. 4x2 – 9 2. 8x5 – 16x4 3. 6xy – 8x – 15y +20

4. 16x4 – 1 5. 3x2 + 17x + 24 6. 6x2 +11x – 10

7. 6x2 – 22x + 24 8. x6 – 1 9. 6x2 – 21x + 15

# 10 – 15: Perform the indicated operations. Show work.

10. 2(x + 3)2 – 3(x – 5) – (x2 – 3x +5) 11. 2x – 6__ x2 – 9 x2 – 9

12. 2x 2 – 11x – 1 2 13. 1 + x + 16 x – 4 4 – x x + 5 x2 – x – 30

14. x 2 – x – 20 ÷ 6x 3 + 24x 2 15. x + 2x_ x2 – 25 2x2 + 10x x – 3 x + 3

1 + 8__ x2 – 9

#16 – 24: Solve algebraically. Show work. # 25 – 27: Solve graphically.

16. 5x – (3x – 4) = 10 – x 17. 1 – 2x < 13

18. x2 + 7x – 30 = 0 19. x2 – 6x + 2 = 0

20. 15x2 – 22x + 8 = 0 21. x3 + 10x2 + 23x = 0

22. x + 3 – 2x – 5 + 1 = x – 1 23. = – 5 5 4 2 4

24. │ x2 + 4x – 3│ = 2 25. x2 – 3x –10 > 0

26. 2x2 + 7x – 4 < 0 27. │ x2 – 4x – 3│ = x3 + x – 6

SEQUENCES and SERIES

# 1 – 15. Show work, using appropriate formulas.

1. Find five arithmetic means between 2. In an arithmetic sequence, the fourth6 and 30. term is 8 and the eighth term is 14.

Find the first term.

3. Find the sum of the multiples of 8 4. Find two geometric means between 2 between 1 and 700. and 54.

5. In an arithmetic sequence, the sixth term 6. In a geometric series, find the first is 12 and the thirteenth term is 40. Find term if r = .5 and the sum of the firstthe common difference and the first term. seven terms is 793.75.

7. Evaluate 8. Express 0.787878… as a fraction.

9. Write 9 + 16 + 23 + 30 + … + 709 10. Find three geometric means betweenin summation (sigma) notation. 3 and 48. (two solutions)

11. Find the 25th term of the sequence 12. In the sequence 16, 8, 4, 2, 1, …b, c, 2c – b, 3c – 2b, … which term is 1/ 256?

13. Find the sum of the first ten terms 14. Find the “sum” of the infiniteof the sequence 4, –12, 36, –108, …. geometric series 81 + 27 + 9 + ….

15. In an infinite geometric series, find the

first term if r = –.25 and S∞ = 60.

LINEAR EQUATIONS and their GRAPHS

# 1 – 8: Find an equation of the line

1. slope = 5, y-intercept = 1 2. x-intercept = 4, y-intercept = 1

3. passing through (3, 9) and (3, – 4) 4. passing through ( – 5, 2) and ( 1, 2)

5. passing through (– 6, 2) and (–1, 12) 6. passing through (– 1, 2) and ( 0, 3)

7. passing through ( 2, 11) and parallel to 8. passing through ( 2, 4) and y = 4x – 2. perpendicular to y = 4x – 2.

# 9 – 10: Find a) slope, b) y-intercept,

c) x-intercept.

9. 2x + 3y = 12 10. x – 4y = 20

# 11 – 12: Find k if the given line contains the given point.

11. – 3x + ky = 24, (4, 6) 12. 5x + 6y = k, (– 2, 5)

POLYNOMIAL FUNCTIONS and RATIONAL FUNCTIONS

Show work.

1. Divide (8x3 + 14x2 – 5x – 9) by (2x + 3). 2. Use synthetic division to divide(3x4 – 8x2 – 11x + 1) by (x –2).

3. Without using division, find the 4. Without using division, determineremainder when the polynomial if x – 4 is a factor of the polynomialx5 + 3x4 + 2x3 + 5x2 + 2x + 17 is 3x3 – 20x2 + 34x – 8.divided by (x + 3).

5. Find all exact real zeros of the function. 6. Find all complex zeros of the functionf(x) = x4– 3x3 – 33x2 – 9x + 44. f(x) = x3 – 8x2 + 37x – 50.(Hint: Begin by graphing the function.)

# 7 – 8: State the a) equation(s) of the vertical asymptote(s), b) equation of the horizontal asymptote, c) coordinates of the hole, if any.

7. y = ____x 2 _______ 8. y = 2x 2 + 8x___ x2 – 9 x2 – 2x – 24

FUNCTIONS and their GRAPHS ______1. f (x) = √ x2 + 9

__ Find a) f (4) = b) f (– 4 ) = c) f ( 6√ 2 ) =

# 2 – 9: Find a) Domain, b) Range

2. y = 3. y = x2 – x – 6

______ _______4. y = √ x2 – 4 5. y = √ 4 – x2

6. y = __x 2 _____ 7. y = │ x – 2│ x2 – 25

_____ ______8. y = √ x – 9 9. y = √ 9 – x2

# 10 – 11 Find a) the coordinates of the vertex of the parabola, b) the y – intercept, c) the x– intercept(s), d) the equation of the axis of symmetry.

10. y = x2 – 4x – 12 11. y = (x + 1)2 – 49

# 12 – 14: Determine whether the function is odd, even, or neither and state what kind of symmetry the function has.

12. y = 2x3 – 4x 13. y = 6x 14. y = 3x2 + 5

# 15 – 18: Given: f(x) = x2 – 4x +1 and g(x) = 3x + 2. Find each of the following. Show work.

15. (f ° g) (5) 16. (g ° f) (5)

17. (f ° g) (x) 18. (g ° f) (x)

# 19 – 20: Find the equation for the inverse of the given function._____

19. f(x) = 3x –12 20. f(x) = √ x – 3

21. Find the average rate of change of 22. If f(x) = x2 + 5x –12, find the f(x) = 2x2 – 3x + 12 from difference quotient.

x = 1 to x = 4.

COMPLEX NUMBERS

# 1 – 3: Simplify. _____1. √ – 16 2. i80 3. i39

# 4 – 12: Perform the indicated operations. Express answers in standard form (a + bi). Show work in # 8 – 12.

4. (5 + 4i) – (2 – 3i) 5. (5 + 4i)(5 – 4i) 6. (3 + 2i)2

__ __ _7. ( √ 7 + 3i)( √ 7 – 3i) 8. ( 6 – √ 3 i)2 9.. 6 + 8i

i

10. __2i___ 11. __120__ 12. _68__ 4 + i 6 – 2i 5 + 3i

# 13 – 15: Solve over the set of complex numbers. Show work.

13. x2 – 6x + 13 = 0 14. x2 + 8x + 17 = 0 15. 9x2 – 12x + 85 = 0

# 16 – 18: Write a quadratic equation with the given roots. Show work.

16. 3 ± 8i 17. – 4 ± 5i 18. , –

EXPONENTS and RADICALS

Evaluate or simplify without using a calculator. Do not use negative exponents in answers.

1. 64⅔ 2. 27 -⅔ 3. 4.

5. 6. 7. 8.

9. x1/3 ( x + x1/2) 10. + 11. 2 −7 + 1

12. a 12 c 5 13. 12x 3 y −6 z 14. (−2 )0

a– 4c8 18x−6y−18z

15. 16. 146 / (5 −7 )

LOGARITHMS

Write as a single logarithm.

1. 3 logb4 + logbw 2. ⅓ log x − log y 3. 3 ln 2 + 2 ln 3

Given: log a = x, log c = y. Express each of the following in terms of x and/or y.

4. log (a5 c4) 5. log (10,000a) 6. log 7. c

Solve for x. If an answer is not exact, round to 3 decimal places. Show work.

8. 2x−5 = 64 9. log3 x = 5 10. log x = 2.985 11. 5x = 30

12. 32x-4 = 13. 45x−7 = 19 14. 16 x + 3 = 32 15. 32x+1 = 4x

Evaluate (correct to 3 decimal places)

16. log 1089.295 17. ln e99.1 18. log5 50

19. If $2,000 is invested at 2.5% per 20. If $10,000 is invested at 3% peryear, compounded monthly, how year, compounded quarterly, inmuch will be in the account at the how many years will this invest-end of 8 years? ment be worth $18,000?

21. Do #20, if interest is compounded 21. At what annual interest rate, compounded

continuously. daily, would $4,000 grow to be worth$7,000 in 20 years?

TRIGONOMETRY

Show work. In # 1 – 2, write each answer as a fraction. In #3 – 12, round off to threedecimal places unless indicated otherwise.

1. In ΔABC, a = 7, b = 24, and 2. If the terminal side of an angle Ө in standardC = 90°. Find the six trig ratios position passes through (−8, −6), find thefor A. six trig function values of Ө.

3. Find two angles (one positive and 4. a) Change 23° 43′ 50″ to the nearestone negative) that are coterminal thousandth of a degree.with the angle 32° in standardposition.

b) Change 37.625° to DMS form.

In # 5 – 8, ΔABC has C = 90°.

5. A = 28° and b = 102.559 cm. 6. B = 53° and c = 17.458 cm.Find a. Find a.

7. a = 19.2 cm and b = 21.7 cm. 8. a = 16.9 cm and c = 23.2 cm.Find B. Find B.

9. Change 11Π to degrees. 10. Change 315° to radian measure in terms 3 of Π.

11. Find the exact length of an arc of 12. Find the degree measure of the referencea circle whose central angle is 40° angle for each angle in standard position.if the diameter is 54 cm. a) 250° b) 285°

Simplify.

13. cos Ө sec 2 Ө − cos Ө 14. sin 4 Ө − cos 4 Ө sinӨ sin2Ө − cos2Ө

15. 1 + cot 2 Ө 16. sin Ө + tan Ө 1 + tan2 Ө 1 + secӨ

GRAPHS OF TRIG FUNCTIONS

Find a) the domain (radian mode) b) the range

1. y = sin x 2. y = cos x 3. y = tan x

4. y = cot x 5. y = sec x 6. y = csc x

7. Write the equations of the vertical 8. Write the equations of the vertical asymptotes in # 3. asymptotes in #4.

9. Write the equations of the vertical 10. Write the equations of the verticalasymptotes in #5. asymptotes in #6.

# 11 – 13. Write a) the amplitude, if any b) the period (radian mode)

11. y = 4sin(x/3) 12. y = −2 cos(Πx/3) 13. y = 3cot(2 Πx/5)

14. Write the equation of the sine function 15. List all of the transformations thatthat has amplitude, period, phase shift, transform the graph of y = cosx and vertical shift, respectively: into the graph of 6, Π/4, −3, 5. y = −2cos ( 2Π/3(x −3)) −2

LAW of SINES and the LAW OF COSINES and AREA

Solve ΔABC and find the area. Write answers correct to three decimal places. Show work.

1. Given: a = 5 cm, b = 8 cm, c = 11 cm 2. Given: a = 6.5 cm, C = 42°, b = 8.2 cm

3. Given: A = 38°, c = 89.95 cm, B = 59° 4. Given: A = 42°, B = 70°, a = 35.9 cm