This document contains 15 multiple choice and free response questions about sinusoidal functions and their graphs. Key concepts covered include: - Identifying amplitude, period, and phase shift from graphs of sinusoidal functions - Writing equations to represent sinusoidal graphs in terms of sine and cosine - Sketching transformed sinusoidal graphs (shifts, stretches, reflections) - Finding amplitude, period, phase shift, and vertical/horizontal shifts from equations - Relating sinusoidal equations to their real world applications like a roller coaster track

1. 1. The figure below shows the graph of y = sin 3 x . What is the value of x at point
P?
π
a)
3
b) 3π
c) 6π
2π
d)
3
Answer: d)
2. The figure shows the graph of y = 2 sin 4 x . What is the value of x at point P?
a) 2π
π
b)
4
π
c)
2
π
d)
8
Answer: b)
3. Which of the following are the amplitude and the period for the function
1
y = − sin ( 3 x − π ) ?
2
1 −1
a) Amplitude: , Period: 2π b) Amplitude: , Period: 2π
2 2
−1 2π 1 2π
c) Amplitude: , Period: d) Amplitude: , Period:
2 3 2 3
Answer: d)
4. The function y = sin ( kx ) has a period of 6π. What is the value of k?

2. 1
Answer:
3
5. What is the range of the function y = sin θ +1 ?
Answer: [0, 2]
 π
6. What is the range for the graph of y = −2 sin 4 x +  + 1?
 2
Answer: [-1, 3]
7.a) State the amplitude, period, horizontal shift, and vertical shift for the sine
function depicted below.
b) Write an equation to describe the sine function.
c) Write another equation for the same curve based on the cosine function.
(5 marks – 2 for part a); 2 for part b); 1 for part c))
Answers:
a) Amplitude: 2
Period: 12
Horizontal Shift: None
Vertical Shift: up 2
π π
b) y = 2 sin x + 2 c) y = 2 cos ( x − 3) + 2
6 6

3. 8. Given the following graph of f ( x ) ,
sketch the following:
a) Sketch f ( − x ) b) Sketch f (x )
1
c) Sketch − 2 f ( x − 5) d) Sketch
f ( x)
(6 marks – 1 each for a) and b), and 2 each for c) and d).)

4. 9. Write two trigonometric equations for the graph below: one in terms of sine and one
in terms of cosine.
(4 marks)
π   3π 
Note: A (0,− ) ; B (π,− ) ; C ( 2π,− ) ; D 
1 1 1 ,2  ; E  ,−4 
 2   2 
 π
Answer: y = 3 sin x −1 and y = 3 cos x −  −1
 2
(Other equations are possible.)
10. Write the equation of the graph sketched below in terms of cos x .
π   3π 
Note: A ,−3  ; B ,3 
4   4 
  3π 
Answer: y = 3 cos 2 x −
   (other equations possible)

  4 

5. 11. Write two equations to represent a sinusoidal function with one maximum at
(2,6), the next maximum at (6,6) and a range of [ − 4,6] . One equation must use a
cosine function and the other a sine function.
(5 marks)
π  π 
Answer: y = 5 cos ( x − 2 )  + 1 and y = 5 sin  ( x − 1)  + 1
2  2 
12.
π   3π 
Point A  ,5  ; Point B  ,−3  ; Point C (π,1)
 4   4 
A math teacher, I. M. Right, told the class that the equation for the graph above was:
  π 
y = −4 cos 2 x +  + 1
  4 
Darcy, the whiz kid, piped up and said the equation should be written as:
  π 
y = 4 cos 2 x −  + 1
  4 
 π
Pat quickly said, “No, no, the equation should be written as: y = 4 cos 2 x −  + 1 .”
 2
a) Who of the above is/are correct? Justify your answer.
b) Lyn said, “I can write an equation for this graph involving the sine function.”
Write such a possible equation.
(5 marks)
Answers: a) all 3 are correct.
b)One possible equation is y = 4 sin 2 x +1 . Other answers possible.

6. 13. On the axes provided, sketch a clearly labeled graph of each of the following, for
at least one period. What is the period of each equation?
π 
a) y = 3cos  (θ − 1)  − 2
3 
  π 
b. y = 2sin  2  θ + ÷ + 2
  2 
14. Write an equation of both sin and cos for the function below.

7. What is the period for the graph?
Show all calculations for a, b, c, and d, where possible.
8
π
-4
15. A portion of a roller coaster track is built in the shape of a sinusoidal curve as
shown below. Find one equation to represent this curve.
Track
28 m
6m
80 m