Functions 11 Exam Review
Chapter
Chapter 8
Section
Functions 11 Exam Review
You need to sign up or log in to purchase.
You need to sign up or log in to purchase.
Lectures 0 Videos
Solutions 96 Videos

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

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

Buy to View
Q1a

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

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

Buy to View
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

Buy to View
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.

Buy to View
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 

Buy to View
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} 

Buy to View
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

Buy to View
Q5

Simplify.

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

Buy to View
Q6a

Simplify.

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

Buy to View
Q6b

Expand. Simplify where possible.

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

Buy to View
Q7a

Expand. Simplify where possible.

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

Buy to View
Q7b

Find a simplified expression for the area of the circle. Buy to View
Q8

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

a) completing the square

b) using a graphing calculator

c) factoring

d) using the quadratic formula

Buy to View
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?

Buy to View
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)

Buy to View
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.

Buy to View
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?

Buy to View
Q13

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} 

Buy to View
Q14b

Simplify and state the restrictions.

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

Buy to View
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} 

Buy to View
Q15b

Simplify and state the restrictions.

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

Buy to View
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 

Buy to View
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 

Buy to View
Q16b

Describe the reflection that transforms f(x) into g(x). Buy to View
Q17a

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

Buy to View
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)

Buy to View
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.

Buy to View
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

Buy to View
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

Buy to View
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

Buy to View
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

Buy to View
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. Find the inverse function and sketch both on the same coordinate system.

Buy to View
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.

Buy to View
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?

Buy to View
Q24

Evaluate without a calculator.

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

Buy to View
Q25a

Evaluate without a calculator.

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

Buy to View
Q25b

Evaluate without a calculator.

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

Buy to View
Q25c

Evaluate without a calculator.

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

Buy to View
Q25cd

Simplify.

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

Buy to View
Q26a

Simplify.

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

Buy to View
Q26b

Simplify.

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

Buy to View
Q26c

Simplify.

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

Buy to View
Q26d

Simplify.

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

Buy to View
Q27a

Simplify.

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

Buy to View
Q27b

Simplify.

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

Buy to View
Q27c

Simplify.

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

Buy to View
Q28a

Simplify.

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

Buy to View
Q28b

Simplify.

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

Buy to View
Q28c

Graph each exponential function. identify the

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

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

Buy to View
Q29a

Graph each exponential function. identify the

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

\displaystyle y =-4^{-x} 

Buy to View
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.

Buy to View
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} 

Buy to View
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 

Buy to View
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}

Buy to View
Q32

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

Buy to View
Q34a

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

Buy to View
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.

Buy to View
Q35

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

Buy to View
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.

Buy to View
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.

Buy to View
Q38

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

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

Buy to View
Q39a

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

\displaystyle \sec A = -6 

Buy to View
Q39b

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

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

Buy to View
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,

Buy to View
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.

Buy to View
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?

Buy to View
Q42

Prove the identity.

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

Buy to View
Q43a

Prove the identity.

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

Buy to View
Q43b

a) Sketch a periodic function, f(x), 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).

Buy to View
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.

Buy to View
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.

Buy to View
Q46

A sinusoidal function has an amplitude of 6 units, a period of 150°, and a maximum at (0, 4).

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

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

Buy to View
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

Buy to View
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.

Buy to View
Q49

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

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

Buy to View
Q50b

Write the first five terms of each sequence.

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

Buy to View
Q51a

Write the first five terms of each sequence.

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

Buy to View
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 5h 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.

Buy to View
Q52

Expand each power of a binomial.

\displaystyle (x -y)^6 

Buy to View
Q53a

Expand each power of a binomial.

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

Buy to View
Q53b

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

a) t_{5, 2}

b) t_{10, 7}

Buy to View
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} 

Buy to View
Q55

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

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

Buy to View
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}, .... 

Buy to View
Q56b

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

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

Buy to View
Q56c

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

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

Buy to View
Q57a

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

\displaystyle a = -5, d = 1.5 

Buy to View
Q57b

Determine the sum of each geometric series.

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

Buy to View
Q58a

Determine the sum of each geometric series.

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

Buy to View
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?

Buy to View
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? Buy to View 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?

Buy to View
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? Buy to View 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?

Buy to View
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?

Buy to View
Q64