Functions Course Review
Chapter
Chapter 7
Section
Functions Course Review
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Solutions 97 Videos

Determine the domain and the range for each relation. Sketch.

\displaystyle y = \frac{3}{x-9}

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Q1a

Determine the domain and the range for each relation. Sketch.

\displaystyle y = \sqrt{2-x} -4

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Q1b

Which of the following is NOT true?

A All functions are also relations.

B The vertical line test is used to determine if the graph of a relation is a function.

C All relations are also functions.

D Some relations are also functions

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Q2

The approximate time for an investment to double can be found using the function n(r) = \frac{72}{r} where n represents the number of years and r represents the annual interest rate, as a percent.

a) How long will it take an investment to double at each rate?

i) 3% ii) 6% iii) 9%

b) Graph the data to illustrate the function.

c) Determine the domain and range in this context.

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Q3

Determine the vertex of each quadratic function by completing the square. Verify your answer by using partial factoring. State if the vertex is a minimum or a maximum.

\displaystyle f(x) =3x^2 + 9x + 1

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Q4a

Determine the vertex of each quadratic function by completing the square. Verify your answer by using partial factoring. State if the vertex is a minimum or a maximum.

\displaystyle f(x) = -\frac{1}{2}x^2 + 3x - \frac{5}{2}

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Q4b

A small company manufactures a total of x items per week. The production cost is modelled by the function C(x) = 50 + 3x. The revenue is given by the function R(x) = 50 + 3x. The revenue is given by the function R(x) = 6x - \frac{x^2}{100}. How many items per should be manufactured to maximize the profit for the company?

Hint: Profit = Revenue - Cost

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Q5

Simplify.

\displaystyle 2\sqrt{243} - 5\sqrt{48}+\sqrt{108}-\sqrt{192}

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Q6a

Simplify.

\displaystyle \frac{2}{3} \sqrt{125} - \frac{1}{3}\sqrt{27} + 2\sqrt{48} -3\sqrt{80}

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Q6b

Expand. Simplify where possible.

\displaystyle (\sqrt{5}+ 2\sqrt{3})(3\sqrt{5} + 4\sqrt{3})

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Q7a

Expand. Simplify where possible.

\displaystyle (4- \sqrt{6})(1 + \sqrt{6})

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Q7b

Find a simplified expression for the area of the circle.

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Q8

Solve 3x^2 + 9x -30 = 0 by

a) completing the square

b) using a graphing calculator

c) factoring

d) using the quadratic formula

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Q9

The length of a rectangle is 5 In more than its width. If the area of the rectangle is 15 m^2, what are the dimensions of the rectangle, to the nearest tenth of a metre?

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Q10

Find an equation for the quadratic function with the given zeros and containing the given point. Express each function in standard form. Graph each function to check.

a) 2 \pm \sqrt{3}, point (4, -6)

b) 4 and -1, point (1, -4)

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Q11

An arch of a highway overpass is in the shape of a parabola. The arch spans a distance of 16m from one side of the road to the other. At a horizontal distance of 1 m from each side of the arch, its height above the road is 6 m.

a) Sketch the quadratic function if the vertex of the parabola is on the y—axis and the road is along the x-axis.

b) Use this information to determine the equation of the function that models the arch.

c) Find the maximum height of the arch.

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Q12

At a fireworks, the path of tho biggest firework can be modelled using the function f(x) = -0.015x^2 + 2.24x +1.75. where x is the horizontal distance from the launching platform. The profile of a hill. some distance away from the platform. can be modelled with the equation g(x) = 0.7x -83. with all distances in metres. Will the firework reach the hill?

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Q13

Are the two function equal? Show your work.

\displaystyle f(x) =-2(x + 3)^2 + (5x + 1) )

\displaystyle g(x)= -2x^2 -7x -17 )

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Q14a

Are the two function equal? Show your work.

\displaystyle f(x) = \frac{x^2 -2x -15}{x^2 -9x + 20} )

\displaystyle g(x)=\frac{x + 3}{x -4} )

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Q14b

Simplify and state the restrictions.

\displaystyle \frac{-x + 1}{8x} \div \frac{2x -2}{14x^2} )

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Q15

Simplify and state the restrictions.

\displaystyle \frac{x^2 + 5x - 36}{x^2-2x} \div \frac{x^2 + 11x + 18}{8x^2 -4x^3} )

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Q15b

Simplify and state the restrictions.

\displaystyle \frac{x^2 -25}{x- 4} \times \frac{x^2 -6x + 8}{3x + 15} )

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Q15c

For each function g(x), state the corresponding base function f(x). Describe the transformations that must be applied to the base function using mapping notation. Then, transform the graph of f(x) to sketch the graph of g(x) and state the domain and range of each function.

\displaystyle g(x) = \frac{1}{x + 5} -1

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Q16a

For each function g(x), state the corresponding base function f(x). Describe the transformations that must be applied to the base function using mapping notation. Then, transform the graph of f(x) to sketch the graph of g(x) and state the domain and range of each function.

\displaystyle g(x) = \sqrt{x + 7} -9

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Q16b

Describe the reflection that transforms f(x) into g(x).

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Q17a

Describe the reflection that transforms f(x) into g(x).

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Q17b

For each of the functions f(x) = x^2, f(x) = \sqrt{x}, and f(x) = \frac{1}{x}, write an equation to represent g(x) and h(x) and describe the transformations. Then, transform the graph of f(x) to sketch graphs of g(x) and h(x) and state the domain and range of the functions.

g(x) = 4f(-x) and h(x) = \frac{1}{4}f(x)

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Q18a

For each of the functions f(x) = x^2, f(x) = \sqrt{x}, and f(x) = \frac{1}{x}, write an equation to represent g(x) and h(x) and describe the transformations. Then, transform the graph of f(x) to sketch graphs of g(x) and h(x) and state the domain and range of the functions.

g(x) = f(4x) and h(x) = -f(\frac{1}{4}x)

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Q18b

A ball is dropped from a height of 32 m. Acceleration due to gravity is -9.8 m/s^2. The height of the ball is given by h(t) = -4.9t^2 + 32.

a) State the domain and range of the function.

b)) Write the equation for the height of the object if it is dropped on a planet with acceleration due to gravity of -11.2 m/s^2.

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Q19

Describe the combination of transformations that must be applied to the base function f(x) to obtain the transformed function g(x). Then, write the corresponding equation and sketch its graph.

f(x) =x, g(x) = -2f[3(x -4)] -1

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Q20a

Describe the combination of transformations that must be applied to the base function f(x) to obtain the transformed function g(x). Then, write the corresponding equation and sketch its graph.

f(x)=\frac{1}{x}, g(x) = \frac{1}{3}f[\frac{1}{4}(x -2)] +3

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Q20b

For each function f(x),

  • i) determine f^{-1}(x).
  • ii) graph f(x) and its inverse.
  • iii) determine whether the inverse of f(x) is a function.

f(x) =4x -5

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Q21a

For each function f(x),

  • i) determine f^{-1}(x).
  • ii) graph f(x) and its inverse.
  • iii) determine whether the inverse of f(x) is a function.

f(x) =3x^2 -12x + 3

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Q21b

The relationship between the area of a circle and its radius can be modelled by the function A(r) =\pi r^2, where A is the area and r is the radius. The graphs of this function and its inverse are shown.

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Q22

A petri dish contains an initial sample of 20 bacteria. After 1 day, the number of bacteria has tripled.

a) Determine the population after each day for 1 week.

b) Write an equation to model this growth.

c) Graph the relation. Is it a function? Explain why or why not.

d) Assuming this trend continues, predict the population after

  • i) 2 weeks
  • ii) 3 weeks

e) Describe the pattern of finite differences for this relationship.

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Q23

Tritium is a substance that is present in radioactive waste. It has a half-life of approximately 12 years. How long will it take for a 50-mg sample of tritium to decay to 10% of its original mass?

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Q24

Evaluate without a calculator.

\displaystyle (-8)^{-2} + 2^{-6}

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Q25a

Evaluate without a calculator.

\displaystyle (3^{-3})^{-2} \div 3^{-5}

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Q25b

Evaluate without a calculator.

\displaystyle (\frac{2^3}{3^2})^{-2}

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Q25c

Evaluate without a calculator.

\displaystyle \frac{6^66^{-3}}{6^2}

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Q25cd

Simplify.

\displaystyle (4n^{-2})(-3n^5)

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Q26a

Simplify.

\displaystyle \frac{12c^{-3}}{15c^{-5}}

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Q26b

Simplify.

\displaystyle (3a^2b^{-2})^{-3}

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Q26c

Simplify.

\displaystyle (\frac{-2p^3}{3q^4})^{-5}

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Q26d

Simplify.

\displaystyle 16^{-\frac{3}{4}}

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Q27a

Simplify.

\displaystyle (\frac{4}{9})^{- \frac{1}{2}}

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Q27b

Simplify.

\displaystyle (- \frac{8}{125})^{- \frac{2}{3}}

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Q27c

Simplify.

\displaystyle \frac{a^{-2}b^3}{a^{\frac{1}{4}}b^{\frac{2}{3}}}

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Q28a

Simplify.

\displaystyle (u^{- \frac{2}{3}}v^{\frac{1}{4}})^{\frac{3}{5}}

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Q28b

Simplify.

\displaystyle w^{\frac{7}{8}} \div w^{- \frac{3}{4}}

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Q28c

Graph each exponential function. identify the

  • domain
  • range-
  • x- and y—intercepts, if they exist
  • asymptote

\displaystyle y = 5(\frac{1}{3})^x

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Q29a

Graph each exponential function. identify the

  • domain
  • range-
  • x- and y—intercepts, if they exist
  • asymptote

\displaystyle y =-4^{-x}

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Q29b

A radioactive sample has a half-life of 1 month. The initial sample has a mass of 300 mg.

a) Write a function to relate the amount remaining, in milligrams, to the time, in months.

b) Restrict the domain of the function so the mathematical model fits the situation it is describing.

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Q30

Sketch the graph of each function, using the graph of y = 8" as the base. Describe the effects, if any, on the

  • asymptote
  • domain
  • range

\displaystyle y = 8^{x -4}

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Q31a

Sketch the graph of each function, using the graph of y = 8" as the base. Describe the effects, if any, on the

  • asymptote
  • domain
  • range

\displaystyle y = 8^{x +2} + 1

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Q31b

Write the equation for the function that results from each transformation applied to the base function y = 11^x.

a) reflect in the X-axis and stretch vertically by a factor of 4

b) reflect in the y—axis and stretch horizontally by a factor of \frac{4}{3}

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Q32

To find trigonometric ratios for 240” using a unit circle, a reference angle of 60° is used. What reference angle should you use to find the trigonometric ratios for 210°?

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Q34a

Use the unit circle to find exact values of the three primary trigonometric ratios for 210° and 240°.

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Q34b

A fishing boat is 15 km south of a lighthouse. A yacht is 15 km west of the same lighthouse.

a) Use trigonometry to find an exact expression for the distance between the two boats.

b) Check your answer using another method.

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Q35

Without using a calculator, determine two angles between 0° and 360° that have a sine of \displaystyle \frac{\sqrt{3}}{2}

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Q36

The point P(-2, 7) is on the terminal' arm of \angle A.

a) Determine the primary trigonometric ratios for \angle A and \angle B, such that \angle B has the same sine as \angle A . b) Use a calculator and a diagram to determine the measures of \angle A and \angle B, to the nearest degree.

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Q37

Consider right APQR with side lengths PQ = 5 cm and QR = 12 cm, and 4Q = 90°.

a) Determine the length of side PR.

b) Determine the six trigonometric ratios for \angle P.

c) Determine the six trigonometric ratios for \angle R.

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Q38

Determine two possible measures between 0° and 360° for each angle, to the nearest degree.

\displaystyle \csc A = \frac{7}{3}

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Q39a

Determine two possible measures between 0° and 360° for each angle, to the nearest degree.

\displaystyle \sec A = -6

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Q39b

Determine two possible measures between 0° and 360° for each angle, to the nearest degree.

\displaystyle \cot C = - \frac{9}{4}

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Q39c

An oak tree, a chest nut tree, and a maple tree form the corners of a triangular play area in a neighbourhood park. The oak tree is 35 In from the chestnut tree. The angle between the maple tree and the chestnut tree from the oak tree is 58°. The angle between the oak tree and the chestnut tree from the maple tree is 49°.

a) Sketch a diagram of this situation. Why is the triangle formed by the trees an oblique triangle?

b) Is it necessary to consider the ambiguous case? Justify your answer.

c) Determine the unknown distances, to the nearest tenth of a metre. If there is more than one possible answer, determine both,

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Q40

At noon, two cars travel away from the intersection of two country roads that meet at a 34° angle. Car A travels along one of the roads at 80 km/h and car B travels along the other road at 100 km/h. Two hours later, both cars spot a jet in the air between them. The angle of depression from the jet to car A is 20° and the distance between the jet and the car is 100 km. Determine the distance between the jet and car B.

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Q41

Isra parks her motorcycle in a lot on the corner of Canal and Main streets. She walks 60 m west to Maple Avenue, turns 40° to the left, and follows Maple Avenue for. 90 m to the office building where she works. From her office window on the 18th floor, she can see her motorcycle in the lot. Each floor in the building is 5 m in height.

a) Sketch a diagram to represent this problem, labelling all given measurements.

b) How far is Isra from her motorcycle, in a direct line?

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Q42

Prove the identity.

\displaystyle \frac{1}{\sin^2\theta} +\frac{1}{\cos^2 \theta} = (\tan \theta + \frac{1}{\tan \theta})^2

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Q43a

Prove the identity.

\displaystyle \csc\theta (\frac{1}{\cot \theta} + \frac{1}{\sec \theta}) = \sec \theta + \cot \theta

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Q43b

a) Sketch a periodic function, fix), with a maximum value of 5, a minimum value of -3, and a period of 4.

b) Determine two other values, b and c, such that f(a) = f(b) = f(c).

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Q44

While visiting a town along the ocean, Bob notices that the water level at the town dock changes during the day as the tides come in and go out. Markings on one of the piles supporting the dock show a high tide of 4.8 m at 6:30 am, a low tide of 0.9 m at 12:40 p.m., and a high tide again at 6:50 pm.

a) Estimate the period of the fluctuation of the water level at the town dock.

b) Estimate the amplitude of the pattern.

c) Predict when the next low tide will occur.

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Q45

Consider the following functions.

\displaystyle \begin{array}{lllll} &(i) \phantom{.} y = 4\sin[\frac{1}{3}(x + 30^o)] -1 \\ &(ii) \phantom{.} y = - \frac{1}{2}\cos[4(x + 135^o)] +2 \end{array}

a) What is the amplitude of each function?

b) What is the period of each function?

c) Describe the phase shift of each function.

d) Describe the vertical shift of each function.

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Q46

A sinusoidal function has an amplitude of 6 units, a period of 150°, and a maximum at (O, 4).

a) Represent the function with an equation using a sine function.

b) Represent the function with an equation using a cosine function.

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Q47

The height, h, in metres, above the ground of a rider on a Ferris wheel after t seconds can be modelled by the sine function h(t) = 9 \sin[2(t - 30)] + 10.

Determine

i) the maximum and minimum heights of the rider above the ground

ii) the height of the rider above the ground after 30 s

iii) the time required for the Ferris wheel to complete one revolution

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Q48

Marcia constructs a model alternating current (AC) generator in physics class and cranks it by hand at 4 revolutions per second. She is able to light up a flashlight bulb that is rated for 6 V. The voltage can be modelled by a sine function of the form y= a\sin[k(x-d)] + c.

a) What is the period of the AC produced by the generator?

b) Determine the value of k.

c) What 1's the amplitude of the voltage function?

d) Model the voltage with a suitably transformed sine function.

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Q49

Write the ninth term. given the explicit formula for the nth term of the sequence.

\displaystyle f(n) = (-3)^{n -2}

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Q50b

Write the first five terms of each sequence.

\displaystyle t_1 =3, t_n = \frac{t_{n-1}}{0.2}

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Q51a

Write the first five terms of each sequence.

\displaystyle f(1) = \frac{2}{5}, f(n) = f(n -1) -1

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Q51b

A hospital patient, recovering from surgery, receives 400 mg of pain medication every 5 h for 3 days. The half-life of the pain medication is approximately 5 h. This means that after 5 h, about half of the medicine is still in the patient’s body.

a) Create a table of values showing the amount of medication remaining in the body after each 5-h period.

b) Write the amount of medication remaining after each 5—h period as a sequence. Write a recursion formula for the sequence.

c) Graph the sequence.

d) Describe What happens to the medicine in the patient’s body over time.

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Q52

Expand each power of a binomial.

\displaystyle (x -y)^6

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Q53a

Expand each power of a binomial.

\displaystyle (\frac{x}{3} - 2x)^4

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Q53b

Write each as the sum of two terms, each in the form t_{n,r}.

a) t_{5, 2}

b) t_{10, 7}

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Q54

State whether or not each sequence is arithmetic. For those sequences that are arithmetic, write the formula for the general term.

\displaystyle \begin{array}{llllll} &i) \phantom{.} 9, 5, 1, -3, ...\\ &ii) \phantom{.} \frac{1}{5}, \frac{3}{5}, 1, \frac{7}{5}, \frac{9}{5}, ...\\ &iii) \phantom{.} -4.2, -3.8, -3.5, -3.3, -3.2, ...\\ \end{array}

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Q55

For each geometric sequence, determine the formula for the general term and then write t_{10}.

\displaystyle 90, 30, 10 ,....

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Q56a

For each geometric sequence, determine the formula for the general term and then write t_{10}.

\displaystyle \frac{1}{4}, \frac{1}{6}, \frac{1}{9}, ....

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Q56b

For each geometric sequence, determine the formula for the general term and then write t_{10}.

\displaystyle -0.0035, 0.035, -0.35, ....

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Q56c

Determine the sum of the first 10 terms of each arithmetic series.

\displaystyle a = 2, d = -3, t_{10} = 56

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Q57a

Determine the sum of the first 10 terms of each arithmetic series.

\displaystyle a = -5, d = 1.5

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Q57b

Determine the sum of each geometric series.

\displaystyle 45 + 15 + 5 + ... + \frac{5}{729}

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Q58a

Determine the sum of each geometric series.

1 -x + x^2 -x^3 + ... - x^{15}

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Q58b

A bouncy ball bounces to \frac{3}{5} of its height when dropped on a hard surface. Suppose the ball is dropped from 45 m.

a) What height will the ball bounce back up to after the seventh bounce?

b) What is the total distance travelled by the ball after 12 bounces?

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Q59

Roy deposits $1000 into a guaranteed investment certificate (GIC) that earns 4.5% per year, simple interest.

a) Develop a linear model to relate the amount in the GIC to time. Identify the fixed part and the variable part.

b) How long will it take, to the nearest month, for the investment to double?

c) What annual rate of interest must be earned so that the investment doubles in 6 years?

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Q60

Suppose you have $2500 to invest for 6 years. Two options are available:

  • Top Bank: earns 6% per year, compounded quarterly
  • Best Credit Union: earns 5.8% per year, compounded weekly

Which investment would you choose and why?

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Q61

Five years ago, money was invested at 6.75% per year, compounded annually.

Today the investment is worth $925.

a) How much money was originally invested?

b) How much interest was earned?

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Q62

At the end of every month, Shivangi deposits $120 into an account that pays 5.25% per year, compounded monthly. She does this for 5 years.

a) Draw a time line to represent this annuity.

b) Determine the amount in the account after 5 years.

c) How much interest will have been earned?

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Q63

Mel plans to withdraw $700 at the end of every 3 months, for 5 years, from an account that earns 7% interest, compounded quarterly.

a) Draw a time line to represent this annuity.

b) Determine the present value of the annuity.

c) How much interest will have been earned?

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Q64