10. f (x)  x2  x  2 is multiplied by k to create a new
function g(x)  kf (x). If the graph of y g(x) passes
through the point (3, 14), state the value of k.
PC 12 Midterm Review
Chapter 1 Review
Multiple Choice
1. y  f (x) contains the point (3, 4). (3, 4) is then
transformed to (5, 5). Which of the following is a
possible equation of the transformed function?
A y  1  f (x  2)
C y  1  f (x  2)
11. Sketch the graph of the inverse relation.
a)
b)
B y  1  f (x  2)
D y  1  f (x  2)
2. y   x  is vertically expanded by a factor of 3 , & then
horizontally translated 3 units left and vertically
translated 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)
12. The graphs of y  f (x) and y  g(x) are shown.
3. y  x is vert. expanded by a factor of 2, reflected
vertically, and then horizontally translated left 3. Which
equation represents the transformed function?
A y  2 x  3
B y  2 x  3
C y  2 x  3
D y  2 x  3
4. Which of the following transformations would produce
a graph with the same x-intercepts a 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)
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).
13.Consider the graph of the function y  f (x).
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 Reflected in the x-axis. B Reflected in the y-axis.
C Reflected on line y  x. D Reflected on 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. When y  f (x) is transformed to y  3f (x  2)  5, the
(2, 5) becomes (x, y). Determine the value of (x, y).
9
1
f (x) is stretched horizontally by a factor of and
2
1
compressed vertically by a factor of . Determine the
3
equation of the transformed function.
a) Describe the transformation of y  f (x) to
y  3f (2 (x  1))  4. b) Sketch the graph.
14. A function is defined by f (x)  (x  2)(x  3).
a) If g(x)  kf (x), describe how k affects the y-int of the
graph yg(x) compared to yf(x)
b) If h(x)  f (mx), describe how m affects the x-int of
the graph of y  h(x) compared to yf(x).
15. Complete the following for f (x)  x2  2x  1.
a) Write the equation in the form y  a(x  h)2 k.
b) What are 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 2 Review
Short Answer
Multiple Choice
1. Which radical function has a domain of
{x | x  2, x  R} & range of { y | y  3, y R} ?
A y 3  x 2
B y3 x2
C y 3 x2
D y3 x2
2. Given that (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-int at 2. If the
graph of the function is stretched horizontally by a factor of
1
2
6. The point (4, y) is on 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 y  x2  25.
8. f (x)  2x is horizontally translated 6 units left. State
the equation of the translated function g (x).
9. This graph is of the function y  f (x).
a) Determine the
equation of the
graph in the form
about the y-axis, what is the new x-int?
A 2
B 1
C
1
D
2
f ( x )  b( x  h)  k .
1
4
b) Determine the
equation in simplest
form.
4. This graph is of the function y  f (x).
10. a) Describe the transformation of y  x to
y  4  2 x  3.
What is the graph of
y
A
f ( x) ?
B
b) State the domain and range of y  4  2 x  3.
c) Explain how the graph of the transformed function
can be used to solve the equation 0  2 x  3  4.
11. f ( x)  x is stretched vertically by a factor of 4,
reflected in the y-axis, vert. translated up 3 units, and
horizontally translated left 5 units. Write the equation of
the transformed function, g (x), and sketch g (x),
12. What number(s) is exactly one third its square root?
C
13. Mary solved x  1  3x  7 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 y  3x  7, and
determined that the point of intersection is (3, 4).
D
a) Determine the correct solution to x  1  3x  7.
b) Explain how Mary’s and John’s solutions relate to
the correct solution.
14. a) Solve 3x  1  2 x 2  2. b) Identify any
restrictions on the variable. c) Verify your solution.
5.
The point (4, 10) is on the graph of the function
f ( x)  k 3( x  1)  4. What is the value of k?
A 2
B 2
C 2
D
1
2
15. On a clear day, the distance to the horizon, d, in
kilometres, is given by d  12.7h , 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?
Chapter 3 Review
Multiple Choice
1. The graph of a third-degree polynomial function of the
form P(x)  ax3  bx2  cx  d 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
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 graph of P(x)ax4 bx3  cx2  dx  e is shown
Consider the following
statements.
i) The y-int at point S is
equal to the constant e.
ii) a  0
iii) The multiplicity of the
zero at point T is 2.
A
B
C
D
Statement i) is true.
Statement ii) is true.
Statement iii) is true.
All statements are true
4. f (x)  (x  4)(x  2)(x  6) is horizontally expanded by a
factor of 2. Which statement 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)  x  7x  kx  17 is divided by
x  5, the remainder is 2. Determine the value of k.
6. The graph of the polynomial function P(x)  a(x  b)(x 
c)(x  d) is shown. Determine the value of a.
3
2
7. If P(x)  x4  bx2  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  6x2  11x  6. The height of the box is x  2. If the
area of the base is 24 cm2, determine the height.
9. Determine the largest possible solution to the
polynomial equation x3  10x2  33x  36.
10. Perform the division (x3  5x2  x  5) ÷ (x  2). Express the
result in the form
P( x)
xa
 Q( x ) 
R
xa
.
11. Factor x4  13x2  12x completely.
12. The graph of y  x3  x2  cx  4 has an x-int of 1.
Determine the value of c and the remaining x-ints
13. Graph f (x)  x3  x2  10x  8. State the x-ints, y-ints,
and the zeros of the function. Determine the intervals
where the function is positive and 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.
Chapter 4 Review
Multiple Choice
1. What is the exact value of csc
A
 2
2
B
2
2
12
5
B
5
12
4
C  2 D
2. Determine tan  if sin  
A
7
C
5
12
?
2
12
and cos   0.
13
D
12
5
3. What are the coordinates of P
  if P() is the point of
7
6
intersection of the terminal arm & the unit circle?
 1  3    3 1   1 3 
  3 1
,  D ,
,  B ,
A 
 C

2
 2
 2 2   2 2  2 2 
4. Solve for  tan2   tan   0 and 0    2.
A
 5
,
4 4
B
3 7 
,
4 4

4
C 0, , ,
5
4
D 0,
3
7
, ,
4
4
5. Determine the general sol. in degrees: 2 cos   1  0
A 240  360n, 300  360n, n  I
B 60  360n, 300  360n, n  I
C 60  360n, 120  360n, n  I
D 120  360n, 240  360n, n  I
12. If sin  
3
, determine the coordinates of P() where
2
the terminal arm of  intersects the unit circle.
 
13. If P() =  3 , 1 , what are the coordinates of P    ?
 2
2
2
14. Consider an angle of
4
radians.
5
a) Draw the angle in standard position.
b) Write a statement defining all angles that are
coterminal with this angle.
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,     .
Short Answer
17. Consider the following trigonometric equations.
6. Convert to radian measure. Answers in exact values.
a) 270
b) –540c) 150
d) 240
A 2 sin   3  0 B 2 cos   1  0
7. Convert the following radian measures to degree
measure. Round answers to 2 decimal places.
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    .
a) 3.25 b) 0.40 c) 
7
d) –5.35
4
8. The minute hand of an analogue clock completes one
revolution in 1 h. Determine the exact value of the
angle, in radians, the minute hand moves in 135 min.
9.
Use the information in each diagram to determine the
value of the variable. Round to 2 decimal places.
a)
b)
C 2 2 sin  cos   2 sin   6 cos   3  0
Chapter 5 Review Multiple Choice
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 equation represents the graph below?
A y  8 sin  x 
1
4 
1
B y  8 sin  x 
2 
c)
C y  8 sin (2x)
D y  8 sin (4x)
d)
3. When y  sin  has been transformed according to the
1


directions y  sin  x   , the horizontal phase shift is
6
2
10. Simplify exactly: sin 2
 
 
5
7
 2 cos (120) tan
.
6
4
11. Given that sin   0.3 and cos   0.5, determine the
value of tan  to the nearest tenth.

units to the right
12

C units to the right
2
A

B

units to the left
2
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. A cyclist pedals at a constant rate of
20 cycles per minute. Write an rquation 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 left, and translated 3 units down.
7. State the amplitude and range for 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
a) Suppose the sound of the boat is modelled by a
sinusoidal function. Which characteristic—amplitude,
period, or range—varies among the three waves?
b) Which parameter in 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?
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?
12. Two sinusoidal functions are shown in the graph.
9.
Consider the graph of y  tan , where  is in radians.
a) What is the general equation of the asymptotes?
b) What are the domain and range?
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) What 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.
Chapter 6 Review
1. Simplify the expression
Multiple Choice
cot 2
.
1  cot 2
A cos2  B sin2 C tan2 
D sec2 
2. The value of (sin x cos x)2  sin 2x is
A 1
B 0
3. The expression
C 1
Answers
D 2
1. D
2. C
Chapter 1 Review Answers
7. { y | y  1, y  R}
1  tan 2
is equivalent to
1  tan 2
4. A
3. C
8. (0, 20)
11. a)
5. B
6. A
1
9. y  f (2 x)
3
10. k  3.5
b)
C cos  D sin 
2
A cos 2 B sin 2
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   cos2  tan2 
C csc   cos  tan  
1  cos2
cos 
D cos2  
tan 
2
Short Answer
b) hor. comp by a factor of
 5π 
6. Determine the exact value of sin    .
 12 
7. Given
sin 2 x
 1.23. What is the value of cos x?
1  cos x
8. If 5  7 sin   2 cos2   0 on the domain 90   
180, what is the value of ?
9. If cosθ 
3π
5
, πθ
, determine the exact value of
13
2
2
10. What single trigonometric function is equivalent to
y
y
sin (3 y) cos    cos(3 y)sin   ?
 2
1
about they-axis; g ( x) 
4
13. a) vertical stretch by a factor of 3 about x-axis, horizontal
stretch by a factor of
1
about the y-axis, reflection in the y-axis,
2
horizontal translation 1 unit right, vertical translation 4 units up
b)
14. a) y-intercept  6k; The original yintercept is multiplied by the value of k.
b) x-intercept =
2 3
m
,
m
; The original x-
intercept is multiplied by the value of
1
m
.15. a) y  (x  1)2 b) (0, 1) c) y  1  x
a) Verify the equation is true for x 
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. g ( x)  2( x  6) 9. a) f ( x)  4( x  1)  2
 2
π
11. Consider the equation sin  x    csc x  1

2
π
.
2
b) Is the equation an identity? Explain.
12. Consider the equation sin2 x  cos4 x  cos2 x  sin4 x.
9b) g ( x)  2 x  1  2 10. a) vertical stretch by a factor of 2 about
the x-axis, translation down 4 units, translation right 3 units
b) domain: {x | x  3, x R} ; range: { y | y  4, y  R}
c) The solutions to 0  2 x  3  4 are the x-intercepts of the
graph of y  2 x  3  4.
11. g ( x)  4 ( x  5)  3
a) Verify the equation for x  30.
b) Prove the equation is an identity.
13. Consider the equation
4x
d) x  1 or x  1
π
sin  θ   .

12. a) vertical exp. by a factor of 2 about the x-axis; g ( x)  2 x
tan x  sec x
sin x

.
cot x
1  sin x
12.
1
9
a) State the non-permissible values if 0  x  360.
b) Prove the equation is an identity algebraically.
14. Solve sin 2x  cos x  0 algebraically if   x  .
15. Solve csc2 x  4cot2 x algebraically. State the general
solution in radians.
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.
1
14. a) x   , x  1 b) There are no restrictions on the variable.
7
15. 83.2 m
Chapter 3 Review Answers
1. C 2. A 3. D 4. A 5. k  7 6. a 
8. 5 cm 9. x  4 10.
x  5x  x  5
3
2
x2
11. x(x  1)(x  3)(x  4)
13.
1
2
7. b  8, c  16
 x 2  3x  7 
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:  
9
x2
12. c  4; x-intercepts: 2, 2
  2
.
, ,
4 3 3
Chapter 5 Review Answers
1. C
 2 
t   19
 3 
2. D 3. D 4. A 5. y  11sin 
1
3
6. y  2 cos ( x  )  3 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 

 n n  I b) domain:{ |  
2

 n  R, n  I} range: { y | y  R} 10. a) period b) b
2
10c) Observer B 11. a)
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 {x | x  R} does not change
under this transformation. The range changes due to the
reflection and the translation; it changes from { y | y  0, y  R}
to { y | y  1, y  R}.
Chapter 4 Review Answers
1. C 2. A 3. A
4. C 5. D 6. a)
4
; 7. a) 186.21°
3
9
2
9. a)   133.69 or 2.33 b) a  31.85 cm
d) a  4.28 ft
10.
c)
5
;
6
b) 22.92° c) 315° d) 306.53°
6d)
8.
3
; b) -3;
2
3
4
1 3
13.  ,

2 2 
14. a)
c) r  6.99 m
 1 3   1 3 
11. 0.6 12.  ,
,  ,

 2 2  2 2 
4
 2n, n  I
5
15. sin   4 , cos   3 ,
5
5

4

5
, csc  
tan  
,
3
4
sec   5 , cot   3
3
4
 
16.  ,
4 4
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) d  0.3 sin
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) f ( x)  6 cos 3x  6, g ( x)  6 cos x  2
Chapter 6 Review Answers
1. A 2. C 3. A 4. B 5. D 6.
9.
 2

; Equation B:  
3 3
4
 6 
2
7. 0.23 8. 150
4
5
 5y
10. sin  
 2
13


11. a) Left side  sin  x  

2
  
 sin   
2
2
b)
17. a) Equation A:   ,
2 
5
2
t    1.1 e) 1.4 s f ) d   0.3 cos
t  1.1;
5 
4
5
 sin   
 0
Right side  csc x  1

 csc  1
2
 11
 0
b) No; it is not true for all permissible values of x.
12. a)
Right side  cos 2 x  sin 4 x
Left side  sin 2 x  cos 4 x
 sin 2 30  cos 4 30
2
 3
 1
    
 2
 2

13
16
4
 cos 2 30  sin 4 30
2
4
 3
 1
   
 2
 2 

13
16
b) Example:
Left side  sin 2 x  cos 4 x
 sin 2 x  (1  sin 2 x) 2
 sin 2 x  1  2sin 2 x  sin 4 x
 1  sin 2 x  sin 4 x
 cos 2 x  sin 4 x
 Right side
Right side  cos 2 x  sin 4 x
13. a) x  0, 90, 180, 270, 360
b) Example:
Left side 
tan x  sec x
cot x
1 
 sin x
 

 cot x
 cos x cos x 
 sin x  1  cos x 
 

 cos x   sin x 
 sin x  1  sin x 
 

 cos x   cos x 



 sin x  1 sin x
1  sin 2 x
 sin x  1 sin x
(1  sin x)(1  sin x)
sin x
(1  sin x)
 Right side
Right side 
sin x
1  sin x
  5
,
2 6 6
14.  ,
15.

2
  n,
  n; n  I
3
3