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EOY Revision Packet
Mathematics, IM3
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Mathematics
IM3
EOY Revision Packet
Ch.1 – Ch.8
Highlights:
 Understand the geometric approach to square a binomial.
 Apply the formula for the square of a binomial.
 Expand powers of polynomials and apply Pascal’s triangle.
 Know and apply the binomial theorem.
 Use the binomial theorem to estimate decimals.
 Use the binomial theorem to evaluate powers of complex numbers.





Divide a polynomial by monomials, binomials, and other polynomials.
Divide using synthetic division.
Know and apply the formula for the sum of geometric series.
Know and apply the remainder theorem.
Know and apply the factor theorem.





Solve a quadratic equation with complex roots. Know and apply the properties of conjugates.
Work on quadratic equations with complex roots.
Apply the fundamental theorem of algebra.
Work out questions on rational roots of polynomial equations.
Graph a cubic and a quartic polynomial function.
 Operate with rational expressions, work out complex rational expressions, and find the zeros of
rational expressions.
 Graph rational functions.
 Solve rational equations.
 Solve rational inequalities algebraically and using a calculator.
 Solve a radical equation and a literal equation.
 Find the inverse relation for different types of expressions.
 Solve a nonlinear equation graphically and numerically using linear interpolation, interval
bisection, and rearranging.
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Mathematics, IM3
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 Express a logarithmic expression in exponential form and an exponential expression in
logarithmic form.
 Evaluate logarithmic expressions from the definition, and estimate the value of a logarithmic
expression.
 Write a log expression in terms of ln.
 Solve a literal equation involving exponentials.
 Solve for the time the formula of amortization.
 Solve for the time the formula of compound interest.
 Work out questions on radioactive material.
 Apply Newton’s law of cooling.
 Graph logarithmic functions and know the inverse property of the logarithms and exponentials.
 Apply the equality and the inequality properties of logarithms.
 Apply the basic properties of the logarithmic function. Apply the product rule and quotient rule of
logarithms.
 Solve simple logarithmic equations and simplify logarithmic expressions.
 Apply the change of base formula.
 Solve exponential equations and inequalities.
 Solve continuous compounding problems, exponential growth and decay problems, loudness of
sound problems, acidity problems, and earthquake intensity problems.




Identify data that is modeled using a cubic polynomial and build a cubic model to represent data.
Use a simple rational function to model data given in a table of values.
Identify data that is modeled using an exponential function.
Use a logarithmic function or a power function to model a set of data.
 Given one algebraic measure of an angle, in degrees or in radians, give other positive/negative
ones.
 Draw a directed angle in a coordinate plane and know the six trigonometric ratios of a directed
angle.
 Know the signs of the trigonometric ratios of directed angles.
 Find the reference angle of a directed angle.
 Know the relations between special trigonometric ratios.
 Express a trigonometric ratio in terms of its reference angle.
 Solve many types of trigonometric equations.
 Draw the graph of the sine function and identify its period, amplitude, vertical shift, and phase
shift.
 Express a sine function as a cosine function and sketch their graphs.
 Sketch graphs of tangent functions.
 Work out different problems on the trigonometric functions.
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EOY Revision Packet
Mathematics, IM3
Page 3 of 14
Vocabulary:

Binomial, Square of a binomial, Pythagorean triplet, Pascal’s triangle, Binomial theorem,
Estimate, Expansion

Divide a polynomial, Factor, Remainder, Synthetic division, Quotient, Divisor

Geometric series, Deposit, Interest, Compounded, Rate, Financed

Remainder theorem, Factor theorem, Linear factors, Multiplicity

Complex roots, Conjugate, Root of a polynomial, Fundamental theorem of algebra

Zeros of a polynomial, Roots of odd multiplicity, Roots of even multiplicities

End behavior, Cubic function, Quartic polynomial

Rational expression, Domain, Simplest form, Least common multiple, Least common
denominator, Simplest form, Complex rational expressions, Asymptotes, Large values

Rational equation, Extraneous solution, Rational inequality, Graphing calculator

Non-linear equation, Radical equation, Literal equation, Inverse of a relation, Graphically,
Numerically, Linear interpolation, Interval bisection, Rearranging

Change of base, Logarithmic equation, Exponential equation, Continuous compounding,
Annual percentage rate, Decay, Growth, Relative intensity, Sound loudness, Acidity of a
substance, Concentration, Magnitude of an earthquake

Modelling, Polynomial function, Rational function, Logarithmic function, Power function,
Exact math model

Trigonometric ratios, Directed angles, Algebraic measure, Remarkable angle, Reference angle,
Related angle, Special angles, Cofunctions, Trigonometric equation, Periodic functions,
Amplitude, Period, Phase shift, Vertical shift

Ferris wheel, Electricity supply, Seasonal sale, Arterial blood flow, Temperature fluctuation
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EOY Revision Packet
Mathematics, IM3
Page 4 of 14
Revision Exercises:
1. Verify the given identity.
a)  2  3    2  3   10
2
2
2
2
3
2
6
b)  x  2x    2x  1  x  1
2. The length of the hypotenuse of a right triangle is 34 units. Find the lengths of the two legs
knowing that they are integers with no common factors.
3. Construct Pascal’s triangle up to row 5 to answer the following questions.
a) What is the coefficient of x3 in the expansion of (2 + x)4?
b) What are the last two terms when (1 + x2)5 is expanded in an ascending order of powers of x?
4
4
c) What is the middle term
1 expansion of (2 + x) ?
 in the
 2

2
d) Expand and simplify 
.
5
 1
t  
e) Expand and simplify  t  .
4. Use the Binomial Theorem to find the indicated terms.
a) The 7th term in the expansion of (u + 2) 11
b) The 5th term in the expansion of (4 – x)8
6
a 

x 
5. The middle term in the expansion of  2 x  is 1. Find the value of a.
6. Simplify (1 + i)5 – (1 – i)3.
7. The total cost function for a product is modeled in dollars by
C(x) = 0.02x2 + 8x + 200, 0 ≤ x ≤ 200.
a) Derive a simplified expression for the average cost function.
b) What is the change in the average cost per unit when the production increases from 10 units to
20 units?
c) What is the change in the average cost per unit when the production increases from 20 units to
30 units?
d) What conclusion can be drawn from the above results?
8. Find the quotient and the remainder using long division and verify your answers.
a) (2x2 +5x + 2) ÷ (x + 4)
b) (2x3 + 3x2 – 6x + 2) ÷ (2x + 1)
9. When the polynomial P(x) is divided by (x + 3), the quotient of the division is –x + 2 and the
remainder is –3. Find P(x).
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Mathematics, IM3
10. Determine P(x) in the synthetic division:
3
1
3
2
P( x)
6
6
2
2
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6
6
11. When 2x2 + ax + 2 is divided by (x + 2), the remainder is 6. Find the value of a.
2 x 2  x3  2 x  3
R
 k  x2 
xa
xa .
12. Consider the division
a) Find the value of a.
b) Write the result of the division.
13. The floor of a rectangular hall is to be covered with rectangular tiles each having a length of
x + 5 cm and a width of 2x + 5 cm. Derive a simplified expression for the minimum number of
tiles needed to cover the whole room given that its area is given by
A = 40x3 + 320x2 + 650x + 250.
2
14. The marketing department expects
monthly
sales of a new product during the first year to be
200tthe
 20
t4
N (t ) 
modeled by the equation
t2 1
, where N is the number of units sold in thousands
and t is the time in months.
200t 2  20t  4
t2 1
a) Perform the division
.
b) Using the result of the division, estimate the maximum monthly sales.
c) Verify your conclusion by calculating N(3), N(4), and N(6).
15. Find the sum of a geometric series given its first term a, its last term an, and the common ratio r.
Express your answer in a simple rational form.
a) a = 1, an = 6,561, r = 3
1
b) a = 1,024, an = 1, r = 4
16. A young worker’s goal is to have $ 250,000 by the time of his retirement. He plans to set aside a
fixed amount at the end of each month in an investment that earns an annual rate of 8%
compounded monthly. If the worker’s intent is to retire after 30 years, what is the amount of
money he should set aside each month?
17. Apply the Remainder Theorem to find P(a).
a) P(x) = 3x3 – 12x2 – 6x + 6, a = 2
b) P(x) = 18 – 10x3 – 12x2 – 20x, a = –3
18. The graph of the equation y = 3x3 + kx2 – 2x – 4 intersects the x-axis at x = – 2. Find k.
19. The remainder of the division of a polynomial P(x) by (4x – 1) is 3. Find P(0.25).
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Mathematics, IM3
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20. Use synthetic division to calculate the value of k that makes the given binomial a factor of the
corresponding polynomial.
a) 2x3 – 5x2 + kx – 15; (x – 3)
b) 18x3 + kx2 – 2x + 1; (2x – 1)
21. Given that (x – 2) and (x + 3) are factors of the polynomial x4 + x3 – 3x2 + ax + b, find a and b
using synthetic division.
22. Show that the polynomial P(x) = 2x4 – 3x3 + 8x2 – 6x + 8 can be expressed as
P(x) = (x2 +2) Q(x). Determine Q(x).
23. Determine the range of values of k for which the roots of the equation are complex.
a) 7x2 – x + k = 0
b) kx2 + 2x + 2 = 0
c) 3kx2 – 6kx + 3k – 1 = 0
24. Evaluate z  z , z  z ,and z z .
a) z = 2i
b) z   3  2i
c)
z
1 i
2
25. Determine the value of k that satisfies the given condition.
a) One root of x2 + kx + 2 = 0 is 1 + i.
b) One root of x2 + kx + 3 = 0 is 1  2i .
26. Find a quadratic equation with real coefficients that has the given complex number as a root.
a) 2  2i
b) 1  i
27. Use the given condition to determine the zeros of the polynomial.
a) x3 + 9x2 + 26x + 24 = 0; P(–2) = 0
b) x3 – 3x2 – 25x + 75 = 0; P(5) = 0
28. Find the cubic polynomial of real coefficients P for which P(2) = P(1 – 2i) = 0
given that P(0) = –10.
29. Given 1 – i is a zero of the polynomial x3 + ax + b, a and b are real numbers. Find a and b and
the other zeros of the polynomial
30. The height of a projectile launched from an elevated hill is given in meters by the equation
h(t) = – t3 + 3t2 + 10t + 40, where t is the time in seconds after its launch.
a) What is the height of the hill of the launch?
b) On its way down, when was the projectile back to its original height (height of the launch)?
c) When was the projectile at a height of 70 meters?
d) For how long was the projectile in the air before hitting the ground?
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Mathematics, IM3
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31. Find the equation of a quartic polynomial P of real coefficients that touches the x-axis at 1
and – 3, and that has a y-intercept of – 27. Use a calculator to verify that the graph of the
polynomial satisfies the given conditions.
32. Find the zeros of the quartic polynomial and then sketch its graph. Whenever applicable,
estimate the local maxima and the local minima, and verify the answers using a calculator.
a) P(x) = x4 – 9x2
b) P(x) = x4 – 2x3 – x2 + 2x
33. The water reserve in a lake during the last year is given in millions of liters by the equation
V(t) = –t3 + 39t2 – 338t + 1040, where t is the number of months from the beginning of the year.
a) What was the volume of the water reserve at the beginning of the year?
b) What was the volume of the water reserve at the end of the year?
c) Sketch the graph of the function V.
d) Estimate the minimum reserve reached during the year.
e) The state of drought is declared in the region when the volume of the water reserve drops
below 200 million liters. Was the state of drought declared last year? If so, during which
months was it declared?
f) Based on the water reserve at the beginning of this year and assuming the pattern of water
consumption is the same as last year, derive an equation for the expected water reserve this
year.
34. State whether the expression is given in its simplest form. If not, simplify the expression and
state the values
excluded from the answer. 3
2
2
( x  3)( x  4)
x  7 x  12 x
a) ( x  2)( x  4 x  3)
2
3
2
b) x  2 x  15 x
35. Perform the indicated operation and express your answer in simplest form.
a)
3x  2 2 x  2

x4
x4
x

36. Perform
the 2division and simplify.
2
x  3x
a) x  3x
2

x 9
x 1
2x
2
x2
2
2

x2  x  2
b) 2 x  5x  2 2 x  3x  2
2
2
2
2 x the complex rational expression.
37. Simplify
x 5
4x
2
x
 25
a)
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3

c) 2 x  x  3 2 x  x  3
b) x  4 x  4
2
a 1  b1
2
2
b) a  b
2y  3
2

4y  6
c) y  2 y  8 y  5 y  4
2
2
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Mathematics, IM3
38. Solve for x.
a)
x 3
4x
0
x2
b) x  4 x  3
2
Page 8 of 14
0
39. The area of a rectangle is 64 cm2. If the length of the rectangle is x cm, derive a simple rational
expression in terms of x to represent its perimeter.
40. A square piece of cardboard of side a feet is used to make a
rectangular box1 open at the top. After removing a small
1
a
square of side a from each corner, the board is folded along
the dotted lines shown in the adjacent figure.
a) Find the area of the base of the box in terms of a.
a
b) Find the total surface area of the box in terms of a.
c) Derive an expression for the ratio of the area of the base to
the total surface area and express it in simplest form.
d) Calculate the ratio in c) for each of the cases where the cardboard has a side of length 2 feet,
3 feet, and 4 feet. Describe the behavior of this ratio as the length of the side increases.
f ( x) 
41. The graph of the function
ax  b
cx  d
has an x-intercept of 8, a y-intercept of – 2, and a vertical
asymptote at x = – 2. Find the values of a, b, and c.
42. A company charges its customers a total cost of 2x + 5 dollars for printing x DVDs.
a) Express the average cost C per DVD as a function in x.
b) Find the equation of the horizontal asymptote of the graph of C.
c) What is the interpretation of this asymptote within the context?
d) What is the average cost per disk when printing 8 DVDs?
e) When does the average cost per DVD drop to $2.50?
43. Solve the equation and verify the answer.
1
a)
1
x

2
x  5 x  25
1
3
5


2
b) x  2 x  1 x  x  2
A
B
x 5


2
44. Find the values of A and B that satisfy the equation x  2 x  1 x  x  2 .
45. Solve the given inequality and use the graphing utility to verify your answer.
a)
x2
0
x3
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b)
x2
4
x2
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Mathematics, IM3
Page 9 of 14
46. The sum of a number and six times its reciprocal is 5. Find the possible values of this number.
47. A company introduces a new toy for the holiday season. They expect sales to reach a maximum
before starting to decrease gradually. The sales
200t s expected in a certain city, in thousands of units
2
sold, are modeled by the equation s(t) = t  100 , where t is the number of days since the toy is
introduced to the market.
a) Use a graphing utility to graph the function.
b) Describe the behavior of the graph.
c) When are the sales expected to reach its maximum and what is the maximum number of toys
to be sold on that day?
d) During which period of time are the sales expected to exceed 6,000 toys?
e) When would the sales drop below 2,000 toys?
f) What happens to the number of toys expected to be sold as t gets very large in value?
48. The two sides of the equation have to be squared twice to remove the radicals. Solve the equation
and check for extreneous solutions.
a)
b) 2a  9  a  8 1
x  1  2 x  14
49. A farmer uses a reservoir of recycled water to irrigate his garden. The rate of water consumption
during an irrigation cycle is modeled by the equation h(t )  6  2 t  3 , where h is the height in feet
of the water in the reservoir, and t is the time in hours since the begining of the cycle.
a) What is the height of the water in the tank at the beginning of the irrigation cycle?
b) How long would it take to empty the reservoir?
c) If the farmer stops the irrigation when the height of the water in the reservoir drops below 50
feet, what would be the length of the irrigation cycle?
d) Use the graphing utility to sketch the graph of h and verify your answers graphically.
e) What are the practical domain and the practical range of the function h?
50. Find the inverse relation E that defines x = E(y).
a)
R( x) 
R( x )  1  x  2
b)
3  2x
3x  1
51. Find the point(s) of intersection of the graphs of f and g by solving the equation f(x) = g(x).
Verify your answer.
2
a) f(x) = 2x + 2, g(x) = 2x + 3x
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b) f(x) = 3
2x + 1
, g(x) =
1
27
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Mathematics, IM3
Page 10 of 14
52. Show that the equation has a root over the given interval. Use two steps of the linear
interpolation to find an approximate value for that root.
3
a) x  2x  5  0;  3  x  2
x
2
b) 2  x  2  0; 1  x  5
53. Show that the equation has a root over the given interval. Use two steps of the bisection method
to find an approximate value for that root.
a) x  x  3  0; 1  x  2
b) x  x  1  0; 1  x  2
54. The cost of buying a laptop that has a cash price of $1,200 is to be paid in weekly installments at
an annual interest rate of 10% compounded weekly.
3
3
2
a) How long would it take to pay off the cost of the laptop and the interest at equal weekly
payments of $135 each?
b) Calculate the finance charges paid.
55. a) The doubling time of a bacteria is 15 minutes. How long would it take this bacteria to triple in
count?
b) Find the doubling time of a bacteria that triples in count every 1.5 hours.
56. Use the graph of y = log x to deduce the graph of y = log (x – 1) + 2. Identify the vertical
asymptote of the graph and state the domain and the range of the function.
57. Use the equality property to solve the equation and check for extraneous solutions.
a) log3 (m2 – 20) = log2 m
b) ln (–6x) = log (x2 + 9)
58. Solve the inequality and verify the answer graphically using the calculator by graphing the
expressions on each side of the inequality.
a) ln 2x > ln (6 – 2x)
b) log (–2x) ≥ log (x + 6)
59. Simplify the expression and express the answer as a single logarithm.
a) log4 (x2 – 4) – log4 (x + 2)
b) log2 (z3 – z2) – 2logz z
60. The count of bacteria in a lab grown culture is modeled t hours after its preparation by the
equation N = 2002 0.4t.
a) What is the initial count of the bacteria?
b) How long does it take the culture to double in count?
c) When does the count N exceed 2,456?
61. A principal P is invested in an account at an annual interest rate compounded continuously. What
is the corresponding annual interest rate if the investment doubles in 7 years?
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62. The current share price of a certain commodity is $7 and is expected to increase at a weekly rate
of 3%. The share price of a second commodity is $9 and is expected to drop by 2% each week.
After how many weeks will the two commodities have the same share price? Calculate this price.
63. How many times more powerful is a sound of 32 decibels than a sound of 30 decibels?
64. Orange juice has a concentration of hydrogen ions that is 10,000 times the concentration in pure
water. What is the pH level of orange juie?
65. The average life expectancy at birth increased noticeably among Americans during the 70’s. It is
still on the rise, but at a slower pace. The equation E = 73.5 + 1.9 ln (t + 1.7) is used to estimate
the average life expectancy at birth t years after 1970.
a) What was the average life expectancy in 1970?
b) What was the increase in the average life expectancy at birth from 1971 to 1980?
c) What is the expected increase in the average life expectancy at birth from 2011 to 2020?
d) How are the results in a) and b) justified by the properties of the graph of the logarithm?
e) When did the average life expectancy at birth reach 79.3 years?
66. The following scatterplot shows the price of a company share at the beginning of each month of
a given year.
Share Price ($)
60
p
50
40
30
20
Jan
Apr
Jul
Month
Oct
Jan
a) What is the degree of the polynomial suggested to fit the data plotted? Justify your choice.
b) Use a calculator to find the equation defining a cubic polynomial function that models the
graph.
c) Verify the validity of the model developed by comparing the calculated share price to the one
deduced from the graph for different years within the given range.
d) Use the graph of the model to estimate the maximum share price over the given data range.
e) Use the graph of the model to estimate the minimum share price over the given data range.
f) Assuming the validity of the model in the following year, what is the expected share price at the
beginning of February of the following year?
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67. The yearly depreciation, D, of a washer depends on its assumed lifetime n. The following table
lists the values of D for different values of n.
n (years)
2
3
4
5
6
D ($)
180
120
90
72
60
a) Produce a scatterplot for the data presented in the table.
b) What model is suggested to represent the given data?
c) Use a calculator to define a function that models the relation between n and D.
d) What does the constant in the equation defining the function represent?
68. The rate of ventilation of a closed space is the time required to exchange the air in a space. The
rate of ventilation of a hall depends on the number of people in this hall. The following table
gives this rate for different numbers of people.
Number of
persons in the
room n
Rate of
ventilation P
(hours)
100
3
200
300
500
700
2.12 1.60 0.95 0.52
a) Draw a scatterplot of the data in the table.
b) Comment on the rate of change of n with respect to P as n increases.
c) Suggest a function to model the data in the table.
d) The function given by n = a + b ln P is suggested to model the relationship between n and P.
Estimate the value of a and b.
e) What is the practical domain of the function?
f) What is the practical interpretation of the domain within the context?
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69. Due to the implemented economic reforms, the financial deficit of a country is declining by the
year. The following table shows the deficit, in billions of dollars, during the period from
2005 – 2012.
Year
Deficit
(billions of
dollars)
2005 2006
2007
180
145.8 131.22 118.1 106.3 95.66 86.09
162
2008
2009
2010
2011
2012
a) Draw a scatterplot of the data.
b) Derive an exponential model to represent the decrease of the deficit along the years.
c) Verify the model by comparing its graph to the scatterplot of the data.
d) What is the average yearly rate of the decrease of the deficit?
e) When will the deficit reach half its value in 2012?
70. A resonant circuit consists of an ideal conductor connected in series with an ideal capacitor. The
resonance frequency of such a circuit is the frequency at which their impedance tends to zero. In
a lab test, a capacitor of capacitance C is connected in series with an inductor of inductance L. The
resonance frequency is a function of the product LC. The following results were obtained to
estimate the resonance frequency f at different products LC. The estimates also depended on the
leakage resistances in the circuit.
LC
10– 6
f (hertz)
172
2 × 10– 3 × 10– 4 × 10– 5 × 10–
6
6
6
6
122
99
86
77
a) Draw a scatterplot of the data.
b) What model do you suggest to represent the data?
c) Use a calculator to generate a power function that models the data.
d) By using a proper scale, compare the graph of the model and the scatterplot of the data.
e) What is the resonant frequency when the value of the product LC is 5 × 10– 5?
71. Express the given radian measure of the angle in degrees.
a)

3
4
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b)
7
6
EOY Revision Packet
Mathematics, IM3
72. To three decimal places, sin

5
= 0.588, cos

5
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= 0.809, and cot

5
= 1.376. Use this information
to evaluate the trigonometric ratios given in 7 – 12. Verify your answers using a calculator.
11π
a) sin 10
b) cos

10
c) tan

10
73. Find the exact values of x between 0 and 2π, in radians, that satisfy the given equation.
a) sin x =

3
2
1
b) cos x = 2
c) tan x =

2
2
74. Use the graph of y = sin x to deduce the graph of the function defined and determine the
amplitude, the period, and the phase shift. Also state the range of the function.
a) y = sin 2x + 0.5
x
2
b) y = 1 + 2 sin
c) y = 2 sin
π

x 
3

+2
75. Use the graph of y = cos x to deduce the graph of the function defined and determine the
amplitude, the period, and the phase shift. Also state the range of the function.
a) y = 1  cos
π

 3x  
3

b) y = 3 cos
 x π
  
4 4
2
76. A consulting firm concluded that the temporary employment in a certain county exhibits a cyclic
behavior throughout the years. The firm suggested a model to simulate the demand on temporary
hiring given by y = 1.8sin(0.8t + 3.5) + 3.2, where y is the number of temporary employees hired
in thousands and t is the time in years since the beginning of 2002.
a) Find the period, the amplitude, the phase shift, and the vertical offset.
b) If the model remains valid until 2020, during which years of the period extending between
2002 and 2020 is the demand for temporary hiring maximum and during which years is it
minimum? Calculate the extreme values for the demand.
c) What is the expected demand in 2015?
d) Sketch the graph of the sinusoidal function over the period between 2002 and 2020 and
describe its features.
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