U5L1- Review of Sinusoidal Graphs and Their Properties From MCF 3M
U5L2-Sinusoidal Graphs Under Transformations
1.
State the phase shift for each trigonometric function.
a) y = sin (x - 60˚)
b) y = cos (x + 90˚)
c) y = sin (x + 30˚) d) y = cos (x - 45˚)
2. Determine the vertical shift and state the range of each function.
a) y = sinx + 3 b) y = cosx – 4
c) y = sinx – 6 d) y = cosx + 5
3. Determine the phase shift and the vertical shift with respect to y = sinx for each function.
a) y = sin(x + 46˚) + 2
b) y = sin(x - 65˚) – 5
4. Determine the phase shift and/or vertical shift and graph one cycle of the function.
a) y = sinx + 1
b) y = cos(x + 30˚)
c) y = sin(x - 60˚) + 2
d) y = cos(x - 120˚) – 1 e) y = sin(x + 45˚) + 3 f) y = cos(x + 90˚) – 2
5. For each graph, determine two equations, one in the form y = cos(x – d) + c and the other in the
form y = sin(x – d) + c.
6. Determine the period for each function.
a) y = 2sinx
b) y = -3cos5x
c) y = cos(-4x)
1
7. Determine the amplitude, period, phase shift, and vertical shift of each function.
amplitude
period
Phase shift
Vertical shift
y  3sin 2  x  60   5
2

y  4cos   x  30   8
5

3
y  sin  3  x  45  
4
y  2cos 9  x  40   5
8. Graph one cycle for each of the following functions.
a) y = sin(2x - 60˚) b) y = 5cos[3(x - 60˚)]
c) y = 3sin(x - 90˚) d) y = cos2x – 1
Answers U5L2
1.a) 60˚ right b) 90˚ left c) 30˚ left d) 45˚ right 2.a) up 3 b) down 4 c) down 6 d) up 5 3.a) left 46˚,
up 2 b) right 65˚, down 5 4.a) no phase shift, up 1 b) left 30˚, no vertical shift c) right 60˚, up 2 d)
right 120˚, down 1 e) left 45˚, up 3 f) left 90˚, down 2 4.
2
Answers U5L2 Con’t
4.
5. a) y = sin(x - 60˚) , y = cos(x - 150˚) b) y = sin(x + 30˚) + 1, y = cos(x - 60˚) + 1
c) y = sin(x - 45˚) – 2, y = cos(x - 135˚) – 2 6.a) k = 2, period = 180˚ b) k = 5, period = 72˚
c) k = -4, period = 90˚ 6. See below
Continued next page
3
Answers U5L2
U5L3 - Applications of Sinusoidal Functions
1. The graph shows the daily high temperature as recorded every 30 days in the town of
Sydenham for two years.
a) Determine the amplitude of the
function.
b) Determine the period of the function.
What does the period represent in this
context?
c) What might be true about a town for
which the graph has a significantly
smaller amplitude?
4
2. The sinusoidal function h  5sin[30(t  4)]  2 models the height, h, in metres, of the tide in a
particular location on a particular day at t hours.
a) Graph one cycle for this function.
b) Determine the maximum and minimum heights of the tide.
c) At what times do high tide and low tide occur?
d) What is the depth of the water at 9:00am?
e) Explain how you could use the graph of this function to model the height of the tide as a
cosine function.
3. The number of students, s, visiting the Centre for Sciences is modeled using the function
s  200sin[30(t  10)]  500 , where t is the time in months since the first of January.
f) Determine the maximum and minimum number of students visiting the Centre for the
Sciences over the period of one year.
g) When is the number of students a maximum? When is it a minimum?
h) How many students visit the Centre for the Sciences on February 14?
i) In what month(s) is the number of students about 600?
j) Suggest a reason for the pattern of attendance represented by this function.
4. A student used a motion detector to gather data on the
motion of a pendulum. She exported the table of values to a
computer and used graphing software to draw the graph.
k) Explain how the graph can be used to determine the
range, amplitude, and vertical shift.
l) Use the graph to determine the period, and then state
the horizontal compression factor, k.
m) Use a sine function to write an equation that models
the motion of the pendulum.
5. The blades on a wind turbine rotate counterclockwise at
30 revolutions per minute. The rotor diameter is 4.8m and the
hub height is 7m.
n) Write an equation to model the height of the tip
on one of the blades above the ground over time. Assume the
blade starts at the point when it is the least distance from the
ground.
o) After how long does this blade reach the
maximum height above the ground?
5
Answers U5L3
1. a) amplitude =
24−(1)
2
= 11.5 b) period = 360 days (The period represents one year.)
1. c) y = -11.5cosx + 12.5 d) There is a smaller change between the maximum and minimum
temperature. The town could be far north or far south where the temperatures are always quite cold
or it could be close to the equator where the temperatures are always quite hot. 2. a)
b) maximum = 7m, minimum = -3m
c) high tide at 7am and 7pm, low tide at 1am and 1pm
d) 4.5m
e) The amplitude, period, and vertical shift would be the same but the phase shift would change.
A possible cosine equation would be h(t) = 5cos[30(t-7)] + 2.
3.
a) maximum = 700, minimum = 300
b) maximum in February, minimum in August
c) 693
d) March, November
e) There is low attendance during the summer when school is not in session.
4.
a) The maximum (4) and the minimum (1) are used to determine the range,
Range = {xεR, 1≤y≤4}.
𝑓
−𝑓
4−1
The amplitude is calculated using the formula, 𝑚𝑎𝑥 2 𝑚𝑖𝑛 = 2 = 1.5.
The vertical shift is the minimum plus the amplitude, 1 + 1.5 = 2.5.
360
b) The period is 3.6. The value of k is 3.6 = 100.
c) The equation is y = 1.5sin(100x) + 2.5.
5.
a) A possible equation is y = 2.4cos[180(x – 1)] + 7.
b) 1 second
6