Math 30-1 Review Package

Name:________________________________ Math 30 – 1 – Final Exam Review Booklet

Chapter 1 (Transformtions) Review

Multiple Choice

For #1 to #6, choose the best answer.

1. The graph y f (x) contains the point (3, 4).

After a transformation, the point (3, 4) is

transformed to (5, 5). Which of the

following is a possible equation of the

transformed function?

A y 1 f (x 2)

B y 1 f (x 2)

C y 1 f (x 2)

D y 1 f (x 2)

2. The graph of y x is transformed by a

vertical stretch by a factor of 3 about the

x-axis, and then a horizontal translation of

3 units left and a vertical translation up

1 unit. Which of the following points is on

the transformed function?

A (0, 0)

B (1, 3)

C (3, 1)

D (3, 1)

3. The graph of y x is vertically stretched

by a factor of 2 about the x-axis, then

reflected about the y-axis, and then

horizontally translated left 3. What is the

equation of the transformed function?

A 2 3 y x

B 2 3 y x

C 2 3 y x

D 2 3 y x

4. Which of the following transformations

would produce a graph with the same

x-intercepts as y f (x)?

A y f (x)

B y f (x)

C y f (x 1)

D y f (x) 1

5. Given the graph of y f (x), what is the

invariant point under the transformation

y f (2x)?

A (1, 0) B (0, 1

2)

C (1, 1) D (3, 1)

6. What will the transformation of the graph of

y f (x) be if y is replaced with y in the

equation y f (x)?

A It will be reflected in the x-axis.

B It will be reflected in the y-axis.

C It will be reflected in the line y x.

D It will be reflected in the line y 1.

Short Answer

7. If the range of function y f (x) is {y y 4},

state the range of the new function

g(x) f (x 2) 3.

8. As a result of the transformation of the

graph of y f (x) into the graph of

y 3f (x 2) 5, the point (2, 5) becomes

point (x, y). Determine the value of (x, y).

9. The graph of f (x) is stretched horizontally

by a factor of 1

2 about the y-axis and then

stretched vertically by a factor of about

the x-axis. Determine the equation of the

transformed function.

10. A function f (x) x2 x 2 is multiplied by

a constant value k to create a new function

g(x) k f (x). If the graph of y g(x) passes

through the point (3, 14), state the value

of k.

1

3

Extended Response

11. Sketch the graph of the inverse relation.

a)

12. The graphs of y f (x) and y g(x) are shown.

13.Consider the graph of the function y f (x).

14. A function is defined by f (x) (x 2)(x 3).

a) If g(x) kf (x), describe how k affects the y-intercept of the graph of the function

y g(x) compared to y f (x).

b) If h(x) f (mx), describe how m affects the x-intercepts of the graph of the function y h(x)

compared to y f (x).

b)

a) If the point (1, 1) on y f (x) maps onto the point (1, 2) on y

g (x), describe the transformation and state the equation

of g (x).

b) If the point (4, 2) on y f (x) maps onto the point (1, 2) on y

g (x), describe the transformation and state the equation

of g (x).

a) Describe the transformation of

y f (x) to y 3f (2 (x 1)) 4.

b) Sketch the graph.

15. Complete the following for the quadratic function f (x) x2 2x 1.

a) Write the equation of f(x) in the form

y a(x h)2 k.

b) Determine the coordinates of the vertex of x f ( y).

c) State the equation of the inverse.

d) Restrict the domain of y f (x) so that its inverse is a function.

Chapter 1 Review - Answers

1. D 2. C

3. A 4. A 5. B 6. A

7. { y | y 1, y R}

8. (0, 20)

9. 1

3(2 )y f x

10. k 3.5

11. a) b)

12. a) vertical stretch by a factor of 2 about the

x-axis; ( ) 2g x x

b) horizontal stretch by a factor of 1

4 about the

y-axis; ( ) 4g x x

13. a) vertical stretch by a factor of 3 about x-axis, horizontal stretch by a factor of 1

2 about the y-axis, reflection in

the y-axis, horizontal translation 1 unit right, vertical translation 4 units up

b)

14. a) y-intercept 6k; The original y-intercept is multiplied by the value of k.

b)2 3

-intercept = ,

m mx ; The original x-intercept is multiplied by the value of

1

m.

15. a) y (x 1)2

b) (0, 1) c) 1y x

d) x 1 or x 1

Chapter 2 (Radical Functions) Review

Multiple Choice


1. Which radical function has a domain of

{ | 2, R}x x x and range of

{ | 3, R} y y y ?

A 3 2y x

B 3 2y x

C 3 2y x

D 3 2y x

2. Given that the point (x, 4x2), x 0, is on the

function y f (x), which of the following is

the point ( )y f x on?

A ( ,x 4x2)

B (x, 2x)

C (x, 2x2)

D ( ,x 2x)

3. The radical function ( )y f x has an

x-intercept at 2. If the graph of the function is

stretched horizontally by a factor of 1

2 about

the y-axis, what is the new x-intercept?

A 2 B 1

C 1

2 D

1

4

4. This graph is of the function y f (x).

What is the graph of ( )y f x ?

A

What is the graph of ( )y f x ?

A

B

C

D

5. The point (4, 10) is on the graph of the

function ( ) 3( 1) 4. f x k x What is the

value of k?

A 2 B 2 C 2 D

Short Answer

6. The point (4, y) is on the graph of

( ) .f x x The graph is transformed into

g (x) by a horizontal stretch by a factor of 2,

a reflection about the x-axis, and a

translation up 3 units. Determine the

coordinates of the corresponding point on

the graph of g (x).

7. State the invariant point(s) when y x2 25

is transformed into 2 25. y x

8. The graph of is horizontally

translated 6 units left. State the equation of

the translated function g (x).

Extended Response

9. This graph is of the function y f (x).

a) Determine the equation of the graph in

the form ( ) ( ) . f x b x h k

b) Determine the equation in simplest form.

10. a) Describe the transformation of y x

to 4 2 3. y x

b) State the domain and range of the

transformed function.

c) Explain how the graph of the

transformed function can be used to

solve the equation 0 2 3 4. x

11. The graph of xxf )( is stretched

vertically by a factor of 4, reflected in the

y-axis, vertically translated up 3 units, and

horizontally translated left 5 units. Write the

equation of the transformed function, g (x),

and sketch the graph.

12. What real number(s) is exactly one third its

square root?

13. Mary solved the radical equation

1 3 7x x algebraically and

determined that the solution is x 3 and

x 2. John solved the same equation

graphically. He sketched graphs of the

functions y x 1 and and

determined that the point of intersection

is (3, 4).

a) Determine the correct solution to the

equation 1 3 7. x x

b) Explain how Mary’s and John’s

solutions relate to the correct solution.

14. a) Solve 23 1 2 2. x x

b) Identify any restrictions on the variable.

c) Verify your solution.

15. On a clear day, the distance to the horizon,

d, in kilometres, is given by 12.7 ,d h

where h is the height above ground, in

metres, from which the horizon is viewed.

If you can see a distance of 32.5 km from

the roof of a building, how tall is the

building, to the nearest tenth of a metre?

1

2

Chapter 2 – Review – Answers

1. A

2. B

3. B

4. D

5. B

6. (8, y 3) or (8, 1)

7. (5, 0), (5, 0)

8. ( ) 2( 6)g x x

9. a) ( ) 4( 1) 2f x x b) ( ) 2 1 2g x x

10. a) vertical stretch by a factor of 2 about the

x-axis, translation down 4 units, translation right

3 units

b) domain: { | 3, R}x x x ;

range: { | 4, R}y y y

c) The solutions to 0 2 3 4x are the

x-intercepts of the graph of 2 3 4. y x

11. ( ) 4 ( 5) 3g x x

12. 1

9

13. a) x 3 b) Example: Since Mary used an algebraic method, she must verify her answers. Only x 3 is a

solution. John determined the point of intersection, but only the x-coordinate of the point of intersection is the

solution.

14. a) x 1

,7

x 1 b) There are no restrictions on the variable.

15. 83.2 m

Chapter 3 Polynomial Functions Review

Multiple Choice

For #1 to # 4, choose the best answer.

1. The partial graph of a third-degree polynomial function of the form

P(x) ax3 bx

2 cx d is shown.

2. Which polynomial function has zeros of 3, 1, and 2, and y-intercept 6?

A (x 3)(x 1)2(x 2) B (x 3)(x 1)(x 2)

C (x 3)(x 1)(x 2) D (x 3)(x 1)(x 2)2

3. The partial graph of the function

P(x) ax4 bx

3 cx

2 dx e is shown.

Which statement about the values of a and d is correct?

A a 0 and d 0

B a 0 and d 0

C a 0 and d 0

D a 0 and d 0

Consider the following statements.

i) The y-intercept at point S is equal to the constant

e.

ii) a 0

iii) The multiplicity of the zero at point

T is 2.

A Only statement i) is true.

B Only statement ii) is true.

C Only statement iii) is true.

D All three statements are true.

4. The graph of the function

f (x) (x 4)(x 2)(x 6) is transformed

by a horizontal stretch by a factor of 2.

Which of these statements is true?

A The new zeros of the function are

12, 8, 4.

B The new zeros of the function are

3, 2, 1.

C The new y-intercept is 96.

D The new y-intercept is 24.

Short Answer

5. When f (x) x3 7x

2 kx 17 is divided by

x 5, the remainder is 2. Determine the value of k.

6. The partial graph of the third-degree polynomial function

P(x) a(x b)(x c)(x d) is shown. Determine the value of a.

7. If P(x) x4 bx

2 c, P(1) 9, and

P(3) 25, what are the values of b and c?

8. The volume of a box is represented by the function V(x) x3 6x

2 11x 6. The height of the box is

x 2. If the area of the base is 24 cm2, determine the height

of the box.

9. Determine the largest possible solution to the polynomial equation

x3 10x

2 33x 36.

10. Perform the division (x3 5x

2 x 5) ÷ (x 2). Express the result in the

form ( )

( )

P x R

x a x aQ x .

11. Factor x4 13x

2 12x completely.

12. The graph of y x3 x

2 cx 4 has an

x-intercept of 1. Determine the value of

c and the remaining x-intercepts.

13. Graph the function f (x) x3 x

2 10x 8. State the x-intercepts, y-intercepts, and the zeros of the

function. Determine the intervals where the function is positive and the intervals where the function

is negative.

14. The graph of the function f (x) x3 is translated horizontally to create g(x). If the point (4, 8) is on

g(x), determine the equation of g(x).

15. The function f (x) x4 is horizontally stretched by a factor of

1

2 about the y-axis, reflected in the x-

axis, and translated vertically 1 unit up. Explain how the domain and range of f (x) are changed by the

transformation.


1. C

2. A

3. D

4. A

5. k 7

6.1

2a

7. b 8, c 16

8. 5 cm

9. x 4

10. 3 2

25 5 9

2 23 7

x x x

x xx x

11. x(x 1)(x 3)(x 4)

12. c 4; x-intercepts: 2, 2

13.

x-intercepts: 4, 1, 2; y-intercept: 9; zeros: 4, 1, 2; positive intervals: (4, 1), (2, ); negative intervals: (,4), (1,

2)

14. g(x) (x 2)3

15. The domain { | R}x x does not change under this transformation. The range changes due to the reflection and

the translation; it changes from{ | 0, R}y y y to { | 1, R}.y y y

Chapter 4 Trig Function / Unit Circle Review

Multiple Choice


1. What is the exact value of csc 7

4

?

A 2

2

B

2

2

C 2 D 2

2. Determine tan if 12

13sin

and

cos 0.

A 12

5

B

5

12

C 5

12 D

12

5

3. What are the coordinates of 7

6P

if P() is the point at the intersection of the terminal arm of angle

and the unit circle?

A 3 1

2 2,

B 1 3

2 2,

C 3 1

2 2,

D 1 3

2 2,

4.Suppose tan2 tan 0 and 0 2. What does equal?

A 5

4 4,

B

3 7

4 4,

C 5

4 40, , , D

3 7

4 40, , ,

5. What is the general solution of the equation 2 cos 1 0 in degrees?

A 240 360n, 300 360n, n I

B 60 360n, 300 360n, n I

C 60 360n, 120 360n, n I

D 120 360n, 240 360n, n I

Short Answer

6. Convert to radian measure. State the method you used to arrive at your solution. Use each conversion

method at least once. Give answers as both exact and approximate measures to the nearest hundredth

of a unit.

a) 270 b) –540

c) 150 d) 240

7. Convert the following radian measures to degree measure. State the method you used to arrive at

your solution. Use each conversion method at least once. Give answers as approximate measures to the

nearest hundredth of a unit.

a) 3.25 b) 0.40 c) 7

4

d) –5.35

9. Use the information in each diagram to determine the value of the variable. Give your answers to the

nearest hundredth of

a unit.

a)

b)

10. Determine the exact value of 2 5 7

6 4sin 2 cos (120 ) tan .

11. Given that sin 0.3 and cos 0.5, determine the value of tan to the nearest tenth.

12. If 3

2sin , determine all possible coordinates of P() where the terminal arm of intersects the

unit circle.

13. If 3 1

2 2P( ) = , ,

what are the coordinates of 2P ?

Extended Response

14. Consider an angle of 4

5

radians.

a) Draw the angle in standard position.

b) Write a statement defining all angles that are coterminal with this angle.

c)

d)

15. The point (3a, 4a) is on the terminal arm of an angle in standard position. State the exact value of

the six trigonometric ratios.

16. Solve the equation sec2 2 0, .

17. Consider the following trigonometric equations.

A 2 sin 3 0 B 2 cos 1 0 C 2 2 sin cos 2 sin 6 cos 3 0

a) Solve equations A and B over the domain 0 .

b) Explain how you can use equations A and B to solve equation C, 0 .


1. C 2. A 3. A 4. C 5. D

6. a) Example: unitary method; 3

;2

4.71

b) Example: proportion method; 3; 9.42

c) Example: unit analysis; 5

;6

2.62

d) Example: unitary method; 4

;3

4.19

7. a) Example: proportion method; 186.21°

b) Example: unitary method; 22.92°

c) Example: unit analysis; 315°

d) Example proportion method; 306.53°

8. 9

2

9. a) 133.69 or 2.33 b) a 31.85 cm c) r 6.99 m d) a 4.28 ft

10. 3

4

11. 0.6

12. 1 3

2 2,

, 1 3

2 2,

13. 1 3

2 2,

14. a)

b) 4

52 , I

n n

15. sin 45 , cos 3

5, tan 4

3, csc 5

4 , sec 5

3, cot 3

4

16. 4 4

,

17. a) Equation A: 2

3 3,

; Equation B:

4

b) Equation C is the product of Equation A times Equation B (i.e., AB C). Therefore, the solution to Equation C is

the solutions to A and B: 2

,4 3 3

,

.

Chapter 5 Trigonometric Graphs Review

Multiple Choice

For 1 to 4, select the best answer.

1. The minimum value of the function

f () a cos b( c) d, where a 0,

can be expressed as

A a d B a d c

C d |a| D d a

b

2. Which of the following is the equation

of the sine wave graphed below?

A 1

8 sin4

y x

B 1

8 sin2

y x

C y 8 sin (2x) D y 8 sin (4x)

3. When the graph of y sin has been

transformed according to the directions

1sin ,

6 2y x

the horizontal phase shift of

the resultant graph is

A 12

units to the right

B 2

units to the left

C 2

units to the right

D 3 units to the left

4.Colin is investigating the effect of changing

the values of the parameters

a, b, c, and d in the equation

y a sin b( c) d. He graphed the

function f (x) sin . He then determined

that the transformation that does not

change the x-intercepts is described by

A g () 2 sin

B h () sin 2

C r () sin ( 2)

D s () sin 2

Short Answer

5. The pedals on a bicycle have a maximum

height of 30 cm above the ground and a

minimum height of 8 cm above the

ground. Out for a ride, a cyclist pedals at a

constant rate of 20 cycles per minute.

Write an equation for this periodic

function in the form y a sin (bt) d.

6. Write the equation of a cosine function in

the form y a cos b(x c) d, with an

amplitude of 2, period of 6, phase shift

of units to the left, and translated 3 units

down.

7. State the amplitude and range for the

graph of y 5 sin 3.

8. a) What system of equations can be

solved using the graph below?

b) State one single equation that can be

solved using the graph. Then, give the

general solution to the equation.

9. Consider the graph of y tan , where is

measured in radians.

a) What is the general equation of the

asymptotes of the graph?

b) What are the domain and range of the

graph of the function?

10. A boat is travelling along a narrow river

between two observers, as shown. The

driver and both observers can hear the

boat’s motor, but the sound that each of

them hears is different, depending on their

location in relation to the boat. The observer

in front of the boat hears a higher-pitched

noise than the driver hears. The observer

behind the boat hears a lower-pitched sound

than the driver hears.

a) Suppose the sound of the boat is

modelled by a sinusoidal function.

Which characteristic—amplitude, period,

or range—varies among the three sound

waves?

b) Which parameter in the equation

y a sin bt d would change if all three

functions were graphed?

c) Which observer’s model equation would

have the largest value of the changing

parameter?

Extended Response

11. You are sitting on a pier when you notice

a bottle bobbing in the waves. The bottle

reaches 0.8 m below the pier, before

lowering to 1.4 m below the pier. The bottle

reaches its highest point every 5 s.

a) Sketch and label a graph of the bottle’s

distance below the pier for 15 s. Assume

that at t 0, the bottle is closest to the

bottom of the pier.

b) Determine the period and the amplitude

of the function.

c) Which function would you consider to

be a better model of the situation, sine or

cosine? Explain.

d) Write the equation of the sine function

that models the bottle’s distance below

the pier.

e) You can reach 0.9 m below the pier.

Use your equation to estimate the length

of time, to the nearest tenth of a second,

that the bottle is within your reach during

one cycle.

f ) Write the cosine function for this

situation. Would your answer for part e)

change using this equation? Explain.

12. Two sinusoidal functions are shown in

the graph.

a) Which characteristics of the two graphs

are the same?

b) Which parameters must change to

transform the graph of f (x) to the graph

of g(x)?

c) Determine the equation for each

of the graphs in the form

y a cos b(x c) d.


1. C

2. D

3. D

4. A

5. 2

311sin 19y t

6.1

32 cos ( ) 3y x

7. amplitude: 5; range: {y | –8 y 2, y R}

8. a) y 2cos x and y 1

b) 2cos x 1; x 60° 360n, n I, and

x 300° 360n, n I

9. a) x 2

n n I

b) domain:{ | 2

n R, n I}

range: { y | y R}

10. a) period b) b c) Observer B

11. a)

b) amplitude is 0.3 m, period is 5 s

c) Ensure that answers are accompanied by an explanation. Example: Cosine curve may not have a phase shift if you

consider a negative a value (that is, a reflection in the x-axis).

d) 2 5

5 40.3 sin 1.1

d t

e) 1.4 s

f ) 2

50.3 cos 1.1;

td Both equations model the same graph, so the result of the calculation would be the same.

12. a) amplitude, horizontal phase shift

b) period or b value, and horizontal central axis

or d value

c) ( ) 6 cos 3 6,f x x ( ) 6 cos 2g x x

Chapter 6 Trig Equations and Identities Review

Multiple Choice

For 1 to 5, choose the best answer.

1. Simplify the expression 2

2

cot

1 cot

.

A cos2 B sin

2 C tan

2 D sec

2

2. The value of (sin x cos x)2 sin 2x is

A 1 B 0 C 1 D 2

3. The expression 2

2

1 tan

1 tan

is equivalent to

A cos 2 B sin 2 C cos2 D sin

2

4. If you simplify sin ( x) sin ( x) it is

A 2 B 0 C 2 D not possible

5. Which of the following is not an identity?

A sec cos sin tan B 1 cos2 cos

2 tan

2

C cos

tancsc cos tan

D 2 1 cos2

2cos

Short Answer

6. Determine the exact value of 5π

12sin .

7. Given 2sin

1 cos1.23.

x

x What is the value of cos x?

8. If 5 7 sin 2 cos2 0 on the domain 90 180, what is the value of ?

9. If 5

13cosθ

,

3π

2π θ , determine the exact value of .

π

2sin θ

10. What single trigonometric function is equivalent to 2 2

sin (3 ) cos cos(3 )sin

y y

y y ?

Extended Response

11. Consider the equation π

2sin csc 1

x x

a) Verify the equation is true for

π

2x . b) Is the equation an identity? Explain.

12. Consider the equation

sin2 x cos

4 x cos

2 x sin

4 x.

a) Verify the equation for x 30. b) Prove the equation is an identity.

13. Consider the equation tan sec sin

cot 1 sin

x x x

x x.

a) State the non-permissible values on the domain 0 x 360.

b) Prove the equation is an identity algebraically.

14. Solve sin 2x cos x 0 algebraically for the domain x .

15. Solve csc2 x 4 cot

2 x algebraically. State the general solution in radians.


1. A 2. C 3. A 4. B 5. D 6. 6 2

4

7. 0.23 8. 150

9. 5

13

10. 5

sin2

y

11. a)

Right side csc 1

csc 12

1 1

0

x

12. a) 2 4

2 4

2 4

Right side cos sin

cos 30 sin 30

3 1

2 2

13

16

x x

13. a) x 0, 90, 180, 270, 360

b) Example:

14. 5

2 6 6, ,

15. 2

, ;3 3

n n n I

Left side sin2

sin2 2

sin

0

x

42

2 4

2 4

1 3

2 2

13

16

Left side sin cos

sin 30 cos 30

x x

b) No; it is not true for all permissible

values of x.

Chapter 7 Exponential Functions Review

Multiple Choice


1. What is the y-intercept for the graph of

y bx 2

, b > 1?

A 2

1

b

B b

2

C 2

1

b D 2

2. In the equation y bx, b > 1, x is replaced by

x 3 and y is replaced by y 4. Which of

the following statements describes the

transformation?

A The point (x, y) on the graph of y bx has

been transformed to the point (x 3, y 4).

B The point (x, y) on the graph of y bx has

been transformed to the point (x 3, y 4).

C The graph of y bx has been translated

4 units to the right and 3 units up.

D The graph of y bx has been translated

3 units to the left and 4 units down.

3. The graph of f(x) ax, a > 1, is transformed

into g(x) 4ax 3

2. Which characteristic

remains the same?

A domain

B range

C x-intercept

D y-intercept

4. The graph of the function f(x) 3ax 2,

a > 0, has the same horizontal asymptote as

which of the following?

A y f(x) 4

B y f(x) 2

C y f(x) 2

D y f(x) 4

5. Mary was asked to solve for x and y in the

exponential equations 5x 3y

1 and

1

525x y . Which of the following linear

equations would lead to a correct solution?

A x 3y 1, x y 1

B x 3y 0, 2(x y) 1

C x 3y 1, 2x y 1

D x 3y 0, x y 2

6. Which function(s) would you graph to solve

the equation 1 4 3

2 1

216

xx

graphically.

A y1 160.5x

, y2 0.54x 3

B 1 4 3

21

1

216

x

xy

C 14 3

21

1

216

xx

y

D y1 4x, 4 3

2

1

2

x

y

Short Answer

7. Given the function f(x) 2x, match the graph

with the correction equation.

a) y f(x) b) y f(x)

c) y f 1

(x) d) y f(x)

I

II

III

8. The function f(x) 5(2x) is transformed by a translation 2 units right and 5 units down. The transformed

function passes through the point (x, 10). Determine the value of x.

9. What vertical translation would be applied to y 4(3x) so that the translation image passes through

(2, 37)?

10. Solve for x.

a) 23 81 3x

b) 2

9 64

16 27

xx

Extended Response

11. You are given the functions y 2x

and

y 2(2x

) 3.

a) Sketch the graphs of the functions on the same grid.

b) Describe the transformation from

y 2x

to y 2(2x

) 3.

c) State the range and the equation of the horizontal asymptote for each function.

d) Determine the value of y when x 400 for each function. Explain how these results relate to your

answers to part c).

12. Consider the graph of the functions f and g.

IV

a) Determine the equation of the

transformed function g(x).

b) Describe the transformation of f(x) to

g(x).

c) Use the graphs to solve the

equation f(x) g(x), to the nearest

hundredth.

13. A single cell of the bacterium E. coli would, under ideal circumstances, divide every 20 minutes.

a) If a culture begins with 1 bacterium, write the equation for the number of bacteria after n

minutes.

b) Determine, to the nearest minute, the time it takes for the culture to grow to 1024 bacteria.

c) If each bacterium has a mass of roughly 1012

g, what is the mass of the bacteria after 1 day, to the

nearest kg?

14. A town had a population of 2200 people in 1990. Each year the population has decreased by 10%.

a) Write an equation to represent the population of the town.

b) What will the population be in the year 2020?

c) When will the population be less than 50 people?

Chapter 7 – Review – Answers 1. C 2. A 3. A 4. D 5. B 6. A

7. a) III b) I c) II d) IV

8. x = 2

9. vertical translation up 1 unit

10. a) 9 b) 4

11. a)

b) vertical stretch by a factor of 2 about the x-axis, and a vertical translation down 3

c) y 2x

: range is y 0, horizontal asymptote is y 0;

y 2(2x

) 3: range is y 3, horizontal asymptote is y 3 d) When x 400, y 2

400 0 and

y 2(2400

) 3 3, both of which correspond to each function's horizontal asympote. These values of x are so large that the y-values are extremely close to the same value as the horizontal asymptote. However, the calculator rounds off the value.

12. a) g(x) 2x 4

2

b) horizontal translation right 4, vertical translation up 2

c) x 1.09

13. a) 201 2

n

A , where A is the number of bacteria, and n is the time, in minutes.

b) 200 min

c) 4 722 366 kg

14. a) P 2200(0.9)n, where n years since 1990 and P population

b) 93

c) 36 years after 1990: 2026

Chapter 8 Logarithmic Functions Review

Multiple Choice

For #1 to 6, select the best answer.

1. The graph of f (x) logb x, b > 1, is translated such that the equation of the new graph is expressed as

y 2 f (x 1). The domain of the new function is

A {x | x > 0, x R} B {x | x > 1, x R} C {x | x > 2, x R} D {x | x > 3, x R}

2. The x-intercept of the function f (x) log5 x 3 is

A 53

B 0 C 1 D 53

3. The equation 2

1

3logy x can also be written as

A 32x

y B 32y

x C 23x

y D 23y

x

4. The range of the inverse function, f 1

, of

f (x) log4 x, is

A { y | y > 0, y R} B { y | y < 0, y R} C { y | y ≥ 0, y R} D { y | y R}

5. A graph of the function y log3 x is transformed. The image of the point (3, 1) is (6, 3). The

equation of the transformed function is

A y 3 log3 (x 3) B y 3 log3 (x 3) C y 3 log3 (x 3) D y 3 log3 (x 3)

6. If log27 x y, then log9 x equals

A 3

2

y B

2

3

y C 3y D 4

y

Short Answer

7. If log3 5 x, express 3 3

1

22log 45 log 225 in terms of x.

8. Determine the value of x algebraically.

a) log4 x 3 b) 2

3log 64

x c) 5log 255 x

d) log3 (x 1)2 2 e) log2 (logx 256) 3

9. Solve for x.

a) log (2x 3) log (x 2) log (2x 1)

b) log (x 7) log (x 3) log (2x 1)

c) 2 log2 (x 4) log2 x 1

10. The point (6, 4) lies on the graph of y logb x. Determine the value of b to the nearest tenth.

Extended Response

11. Solve the equation 5x 104, graphically and algebraically. Round your answer to the nearest

hundredth.

12. Given f (x) log3 x and g(x) log3 9x.

a) Describe the transformation of f (x) required to obtain g(x) as a stretch.

b) Describe the transformation of f (x) required to obtain g(x) as a translation.

c) Determine the x-intercept of f (x). How can the x-intercept of g(x) be determined using your answer

to parts a) or b)?

13. Explain how the graph of 4log (3 1)

21

xy can be generated by transforming the graph of

y log4 x.

14. Identify the following characteristics of the graph of the function y 2 log4 (x 1) 3

a) the equation of the asymptote

b) the domain and range

c) the x-intercept and the y-intercept

15. An investment of $2000 pays interest at a rate of 3.5% per year. Determine the number of months required for the investment to grow to at least $3000 if interest is compounded monthly.

16. Radioactive iodine-131 has a half-life of 8.1 days. How long does it take for the level of radiation to reduce to 1% of the original level? Express your answer to the nearest tenth.

Chapter 8 – Review – Solutions 1. B

2. A

3. D

4. A

5. A

6. A

7. x 3

8. a) 1

64 b) 512 c) 25 d) 4, 2 e) 2

9. a) 3.5 b) no solution c) 8

10. 0.6

11. 2.89

12. a) horizontal stretch by a factor of 1

9 about the y-axis

b) vertical translation 2 units up

c) x-intercept of f (x) is 1; the x-intercept of g(x) is 1

9, since g(x) is a result of a horizontal stretch by a factor of

1

9

13. vertical stretch by a factor of 1

2 about the x-axis, a horizontal stretch by a factor of

1

3 about the y-axis, a

horizontal translation 1

3 units right, and a vertical translation 1 unit up

14. a) x 1 b) domain: {x x 1, x R}; range: { y y R} c) x 7

8 , y 3

15. 140 months

16. 53.8 days

Chapter 9 Radical Functions Review

Multiple Choice


1. The x-intercept of 1

2

k

xy is 0.5. What

is the value of k?

A 1.0 B 1.5

C 2.5 D 3.0

2. Consider the function 2

2

1( )

x

xg x . Which

statement is false?

A g(x) has two vertical asymptotes.

B g(x) is not defined when x 0.

C g(x) has one zero.

D g(x) is a rational function.

3. Consider the functions f(x) x x2, g(x) 2x 1,

and ( )

( )( )

f x

g xh x . Which statement is true?

A f(x), g(x), and h(x) have the same domain.

B The zero of f(x) is the vertical asymptote

of h(x).

C The non-permissible value of h(x) is the

zero of g(x).

D h(x) is equivalent to y 0.5x 0.25.

4. Consider the following graph of the function

2 1( )

x

x rf x .

What is the value of r?

A 3 B 2

C 2 D 3

5. Which of the following is true of the

rational function 3

26

xy ?

A It has a zero at x 2.

B Its range is {yy R}.

C It is equivalent to6 9

2

x

xy .

D It has a vertical asymptote at x 6.

6. The graph of which function has a point of

discontinuity at x 1?

A 2

1

1

x

xy B

2

1

1

x

xy

C 2 1

1

x

xy D

2 1

1

x

xy

7. Which function has a domain of

{xx 1, x R} and a range of

{yy 3, y R}?

A 31

x

xy B

2

2

3 3

4 3

x x

x xy

C 3

1

x

xy D

2

2

3

x

x xy

8. How many roots does the equation

2

8 1

16 41

x x have?

A 0 B 1

C 2 D 3

Short Answer

9. a) Sketch the graph of the function

2

2

4

x

xy .

b) Identify the domain, range, and asymptotes of the function.

c) Explain the behaviour of the function as the value of |x| becomes very large.

10. a) Sketch the graph of the function

2

51

xy .

b) State the values of the x-intercept and

y-intercept.

c) Solve 2

50 1

x algebraically.

d) How is your answer to part c) related to your answers to parts a) and b)?

11. Select the graph that matches the given function.

a) 2

2

( 3)5

xy b)

2

23

( 5)

xy c)

2

2

( 5)3

xy

B A C

Chapter 10 Function Operations Review

Multiple Choice

For #1 to #5, select the best answer.

1. From the graph, what is the value of (f g)(2)?

A 3 B 0 C 2 D 4

2. Given f (x) x2 2 and g(x) x 5, which

equation represents h(x) ( f g)(x)?

A h(x) 2x2 5 B h(x) x

2 x 3

C h(x) x2 x 5 D h(x) x

2 2x 5

3. Let f (x) x 1 and g(x) x2 1. Determine

the non-permissible values of ( )

f

gy x .

A 1 B 1 C 1 D none

4. If f (x) 3 1x and g(x) x2, which is the

domain of ( )

( )( )

f x

g xm x ?

A {x | x 0, x R} B {x | x 0, x R}

C 1

3, Rx x x

D 1

3, Rx x x

5. Consider the functions f (x) x2 2 and

g(x) 1 x . Which statement is true?

A ( )

( )0, 1

f x

g xx B f (x) g(x) 0

C f (x) g(x) D (g f ) (x) 1

Short Answer 6. Given f (x) x and g(x) 4 x, match

the combined function in set A with the graph in set B.

Set A

i) ( f g)(x) ii) ( f g)(x)

iii) f(x)g(x) iv) ( )

f

gx

Set B

A.

B.

C.

D.

8. Given the functions f (x) 1

x and

1

1( )

xg x , determine the equation of the combined function h(x).

Then state the domain of h(x).

a) h(x) (f g)(x) b) h(x) (f g)(x) c) h(x) f(x)g(x) d) h(x) ( )

f

gx

9. Let f (x) x 1, g(x) x2 1, and h(x) 1 x. Determine each equation.

a) q(x) f (x) h(x) b) p(x) g(f (x))

10. Find two functions, f (x) and g(x), such that f (g(x)) (2x 3)2 5.

Extended Response

11. Consider the functions f (x) x2 and g(x) 2

x.

a) Determine the equation of ,( )

( )( )

f x

g xh x and state the domain of h(x).

b) How does the graph of h(x) behave for large values of x?

12. Assume f (x) x and g(x) |x|.

a) Determine the equation of 3 ( ) ( )

( )( )

f x g x

f xh x

.

b) Sketch the graph of h(x). c) State the domain and range of h(x).

13. If f (x) x2 and h(x) x 1, then g(x) 3( f (h(x))) 5.

a) Determine an equation for g(x). b) Describe g(x) as a transformation of f (x).

14. Let h(x) cos x and g(x) 1

x. Determine the composite functions h(g(x)) and g(h(x)), the domain


1. D 2. B 3. C 4. C 5. C 6. i) D ii) A iii) B iv) C 7. a) 5 b) 1 c) 1 d) 3

8. a) 2

2 1

x( )

x

xh x

; {x | x 0, 1; x R} b)

2

1( )

x xh x

; {x | x 0, 1; x R} c)

2

1( )

x xh x

; {x | x 0, 1;}

d) 1

( )x

xh x

; {x | x 0, 1; x R} 9. a) q(x) 2 b) p(x) x

2 2x 2 10. f (x) x

2 5; g(x) 2x 3

11. a)

2

2( )

x

xh x ; x R b) as x increases, h(x) approaches 0

12. a) 3 | |

( )x x

xh x

b)

c) domain: {x | x 0, x R};

range: {y | y 2, 4; y R}

13. a) g(x) 3(x 1)2 5 or g(x) 3x

2 6x 2

b) vertically stretched by a factor of 3 about the x-axis, translated left 1

and translation down 5

14. a) 1

( ( )) cosx

h g x

; domain: {x | x 0, x R} b)

( ( )) secg h x x ;domain: 2| , I, R

ππx x n n x

Chapter 11 Permutations and Combinations Review

Multiple Choice

For # 1 to #6, choose the best answer.

1. The number of 3-digit numbers, with repeats,

that are multiples of 5 and less than 600 that

can be formed from the digits 1, 3, 5, 7,

and 9 is

A 6 B 9 C 15 D 25

2. The grid shows the roads from point A to

point C.

If the only allowed directions are south and

east, the number of pathways from point A to

point C that do not pass through point B is

A 9!

5!4!5!4!

B

9!

5!4!5!3!

C 9! 5!

5!4! 4!

D

9! 5!

5!4! 3!

3. The number of different arrangements of the

letters of the word CONFERENCE is

A 10!

3!4!

B 10!

3!

C 10!

2!3!

D10!

2!2!3!

4. If nP3 = 2(nC4), then nC6 equals

A 1716 B 3003 C 5005 D 8008

5. Given that the first 4 terms of a row in

Pascal’s triangle are 1, 9, 36, and 84, which of

the following is equivalent to the middle term

of the next row?

A 10C6 B 2(10C5) C 2(9C6) D 9C4 + 9C5

6. Which of the following statements is always

correct?

A The number of permutations of n different

elements taken r at a time is less than the

number of combinations of n different

elements taken r at a time.

B The number of permutations of n different

elements taken r at a time is greater than

the number of combinations of n different

elements taken r at a time.

C The number of permutations of n different

elements taken r at a time is less than or

equal to the number of combinations of n

different elements taken r at a time.

D The number of permutations of n different

elements taken r at a time is greater than or

equal to the number of combinations of n

different elements taken r at a time.

Short Answer

7. For a graduation ceremony, there are 8 people

sitting on stage in one row. If the master of

ceremonies must sit on the far left and the

valedictorian must sit next to the principal,

how many arrangements are possible?

8. A pizzeria advertises that it has 12 toppings.

How many different pizzas can be made with

0 to 5 toppings? Assume each topping can be

chosen only once per pizza.

9. Assume that a handshake takes 6 s. How

long, in minutes, does it take for all 45

students in a graduating class to shake hands

with each other, if everyone shakes hands

once with every other person in the class?

10. Arrange the following expressions from least

in value to greatest:

7C3, 6P4, 6!

2!4!3

, 14 3

13 2

P

C

11. Solve for n: 75 CC nn

12. Determine the value of the term independent

of x in the expansion of

5

2

3

13

xx .

Continued top of next column >>

Extended Response

13. Mary is packing for her trip. She laid out her

clothes and drew the following diagram.

a) How many different outfits are possible?

b) Mary decides to add one more top and

one more bottom. How many more outfits

are possible?

14. Three students were selected from the

23 grade 12 students to be the president,

vice-president, and secretary of the student

council. Three other students from the

38 grade 10 and grade 11 students were also

selected to be on the council.

a) Determine the total number of ways that

the 6 members of the student council can

be selected.

b) Explain why both permutations and

combinations are needed to answer

part a).

15. Solve for n algebraically: 126(nC3) = n + 1P5.

16. The social justice group is comprised of

4 grade 10 students, 5 grade 11 students, and

6 grade 12 students. Determine how many

ways a committee of 5 students can be

selected if

a) there are no restrictions.

b) there are exactly 2 grade 12 students on

the committee.

c) there are at most 2 grade 12 students on

the committee.

17. Consider the expansion of (x + y)6.

a) State the coefficients in the expansion of

(x + y)6 using Pascal’s triangle.

b) State the coefficients in the expansion of

(x + y)6 using combinations.

c) Use the answer from part a) or part b) to

expand (2x – 3y2)6. Simplify the result.

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Math 30-1 Review Package