5.5 Stretches of Periodic Functions
Light waves can be modelled by sine functions. The
graphs model waves of red and blue light. The units
of the scale on the x-axis are nanometres, where one
nanometre (1 nm) is a billionth of a metre, or 10−9 m.
y
1
0
x = 460
100
200
300
400
x = 670
500
600
700
800
x
–1
Notice that the graphs for red and blue light are
transformations of y = sin x. These graphs will be
used in Example 5.
I NVESTIGATE & I NQUIRE
Copy and complete the table by finding decimal values for sin x and 2sin x.
Round values to the nearest tenth, if necessary.
1.
x (degrees)
sin x
2sin x
0
0
0
30 60 90 120 150 180 210 240 270 300 330 360
0.5
1
Sketch the graphs of the
functions y = sin x and
y = 2sin x on the same grid
like the one shown.
2.
y
2
1
0
60
120
180
240
300
360
x
–1
–2
3.
a)
For the graphs from question 2, find
the amplitudes
b) the periods
4.
What transformation can be applied to y = sin x to give y = 2sin x?
c)
any invariant points
5.5 Stretches of Periodic Functions • MHR 367
Make a conjecture about how the graph of y = 1 sin x compares with
2
the graph of y = sin x.
1
b) Use your conjecture to sketch the graphs of y = sin x and y = sin x on
2
the same grid for the domain 0° ≤ x ≤ 360°.
5. a)
Test your conjecture from question 5a) by graphing y = sin x and
y = 1 sin x, 0° ≤ x ≤ 360°, on a graphing calculator. Compare the result
2
with your graph from question 5b).
6.
7.
a)
For the graphs from question 6, find
the amplitudes
b) the periods
c)
any invariant points
What transformation can be applied to y = sin x to give y = 1 sin x ?
2
9. If a > 0, write a statement describing the transformational effect of a on
the graph of y = asin x.
8.
If a > 0, write a conjecture about the transformational effect of a on the
graph of y = acos x.
10.
Test your conjecture from question 10 by graphing y = cos x,
y = 2cos x, and y = 1 cos x, 0° ≤ x ≤ 360°, on a graphing calculator.
2
12. For the graphs from question 11, find
a) the amplitudes
b) the periods
c) any invariant points
11.
I NVESTIGATE & I NQUIRE
Copy and complete the tables of values for y = sin x and y = sin 2x.
Round values to the nearest tenth, if necessary.
1.
x (degrees)
sin x
sin 2x
0
0
0
30 60 90 120 150 180 210 240 270 300 330 360
0.5
0.9
Sketch the graphs of the functions y = sin x and y = sin 2x on the same
grid.
2.
3.
a)
For the graphs from question 2, find
the amplitudes
b) the periods
4.
What transformation can be applied to y = sin x to give y = sin 2x?
368 MHR • Chapter 5
c)
any invariant points
Make a conjecture about how the graph of y = sin 1x compares
2
with the graph of y = sin x.
1
b) Use your conjecture to sketch the graphs of y = sin x and y = sin x
2
on the same grid for the domain 0° ≤ x ≤ 720°.
5. a)
Test your conjecture from question 5a) by graphing y = sin x and
y = sin 1x, 0° ≤ x ≤ 720°, on a graphing calculator. Compare the result
2
with your graph from question 5b).
6.
7.
a)
For the graphs from question 6, find
the amplitudes
b) the periods
c)
any invariant points
What transformation can be applied to y = sin x to give y = sin 1x?
2
9. If k > 0, write a statement about the transformational effect of k on the
graph of y = sin kx.
8.
If k > 0, write a conjecture about the transformational effect of k on the
graph of y = cos kx.
10.
Test your conjecture from question 10 by graphing y = cos x,
y = cos 2x, and y = cos 1x, 0° ≤ x ≤ 720°, on a graphing calculator.
2
12. For the graphs from question 11, find
a) the amplitudes
b) the periods
c) any invariant points
11.
The transformations that apply to
algebraic functions also apply to
trigonometric functions.
If a > 1, the graphs of y = asin x
and y = acos x are stretched
vertically by a factor of a. The
amplitude of each function is a.
y
3
2
1
0
–1
If 0 < a < 1, the graphs of
y = asin x and y = acos x are
compressed vertically by a factor
of a. The amplitude of each
function is a.
y = 3sin x
y = sin x
60
120
y = 1– sin x
180
240
300
360
x
3
–2
–3
5.5 Stretches of Periodic Functions • MHR 369
The five-point method is a convenient way to sketch the graph of a y
sine or cosine function using its amplitude and period. This method
uses the fact that one cycle of a sine or cosine function includes a
maximum, a minimum, and three zeros. These five key points are
0
equally spaced along the x-axis, so they divide the period into
quarters.
maximum
zero
x
zero
minimum
Suppose the graph of a sine function has an amplitude of 3 and a
period of 2π. Because the five key points divide the period into quarters,
π
3π
the coordinates of the five key points are (0, 0), , 3 , (π, 0), , −3 ,
2
2
and (2π, 0).
To sketch the graph, use scales on the y-axis and x-axis that are
about equal. Use π =⋅ 3. Plot the five key points in the cycle.
Draw a smooth curve through the points.
y π– , 3
2
2
(0, 0)
0
(π, 0)
π
(2π, 0)
x
2π
–2
3π , –3
–––
2
EXAMPLE 1 Sketching y = acos x
Sketch one cycle of the graph of y = 4cos x,
starting at (0, 0), x ≥ 0. State the domain
and range of the cycle.
Note that the transformations shown in this chapter
can be performed using a graphing software program,
such as Zap-a-Graph. For details of how to do this,
refer to the Zap-a-Graph section of Appendix C.
SOLUTION
The graph of y = 4cos x is the graph of y = cos x expanded vertically by a
factor of 4.
The amplitude is 4, so the maximum value is 4 and the minimum value is −4.
The period of the function y = 4cos x is 2π.
Use the five-point method to sketch the graph.
The five key points divide the period into quarters.
Therefore, the coordinates of the five key points are
π
3π
(0, 4), , 0 , (π, −4), , 0 , and (2π, 4).
2
2
Plot the 5 key points in the cycle. Draw a smooth curve through
the points. Label the graph.
The domain of the cycle is 0 ≤ x ≤ 2π.
The range of the cycle is −4 ≤ y ≤ 4.
370 MHR • Chapter 5
y
4
2
0
π
2π
–2
–4
y = 4cos x
x
If k > 1, the graphs of y = sin kx and y = cos kx
are compressed horizontally by a factor of 1.
k
2
π
36
0°
The period of each function is or .
k
k
y
y = sin x
y = sin 4x
1
0
π
4π x
3π
2π
–1
If 0 < k < 1, the graphs of y = sin kx and
y = cos kx are expanded horizontally by a
factor of k. The period of each function is
2π
360°
or .
k
k
y
y = cos x
1
0
–1
π
3π
2π
4π
5π
x
1
y = cos – x
3
Example 2 Sketching y = sin kx
Sketch one cycle of the graph of y = sin 3x, starting at (0, 0), x ≥ 0.
State the domain and range of the cycle.
SOLUTION
The graph of y = sin 3x is the graph of y = sin x compressed horizontally
by a factor of 1.
3
The amplitude is 1, so the maximum value is 1 and the minimum value is −1.
2π .
The period of the function is 3
Use the five-point method to sketch the graph.
The five key points divide the period into quarters.
Therefore, the coordinates of the five key points are
π
π
π
2π
(0, 0), , 1 , , 0 , , −1 , and , 0 .
6
3
2
3
y
Plot the 5 key points in the cycle. Draw a smooth
y = sin 3x
curve through the points. Label the graph.
1
2π
The domain of the cycle is 0 ≤ x ≤ .
x
0
π
3
–1
The range of the cycle is −1 ≤ y ≤ 1.
5.5 Stretches of Periodic Functions • MHR 371
EXAMPLE 3 Sketching a Sine Function
A sine function has an amplitude of 4 and a period of 6π. If one cycle of the
graph begins at the origin, and x ≥ 0, sketch one cycle of this sine function.
SOLUTION
The cycle begins at the origin and the amplitude is 4, so the maximum value
is 4 and the minimum value is −4.
The period of the function is 6π.
Use the five-point method to sketch
the graph. The five key points divide
the period into quarters. Therefore, the
coordinates of the five key points are
3π
9π
(0, 0), , 4 , (3π, 0), , −4 ,
2
2
and (6π, 0).
y
4
2
0
π
2π
3π
4π
5π
–2
–4
Plot the 5 key points in the cycle. Draw a smooth curve through the points.
EXAMPLE 4 Graphing y = acos kx, –π ≤ x ≤ π
Sketch the graph of y = 3cos 2x over the domain −π ≤ x ≤ π.
SOLUTION
The graph of y = 3cos 2x is the graph of y = cos x expanded vertically by
a factor of 3 and compressed horizontally by a factor of 1.
2
The amplitude is 3, so the maximum value is 3 and the minimum value is −3.
2π , or π.
The period of the function is 2
Use the five-point method to sketch the graph for 0 ≤ x ≤. π.
The five key points divide the period into quarters.
Therefore, the coordinates of the five key points are
π
π
3π
(0, 3), , 0 , , −3 , , 0 , and (π, 3).
4
2
4
Plot the 5 key points in the cycle. Draw a smooth curve
through the points.
372 MHR • Chapter 5
y
2
0
–2
π
x
6π x
You can verify the graph using a graphing calculator
in radian mode, with Xmin = 0 and Xmax = π.
Use the pattern to sketch the graph over required
domain. Label the graph.
y y = 3cos 2x
1
–π
0
π
x
–1
EXAMPLE 5 Writing the Equations of Light Waves
The sine graphs model waves of red light
and blue light, where the units of x are
nanometres. Write equations for red light
waves and blue light waves
a) using exact values
b) using approximate values, to the nearest
thousandth
y
1
0
x = 460
100
200
300
400
x = 670
500
600
700
800
x
–1
SOLUTION
The amplitude for both colours is 1.
From the graph for red light, one cycle
has a length of 670 nm, so the period is 670.
2π
Period = k
2π
670 = k
2π or π
k=
670
335
The equation for red light waves
x , using exact values.
is y = sin π
335
a)
b)
In the equation for red light,
k = π
335
=⋅ 0.009
The approximate equation for red
light waves is y = sin 0.009x.
From the graph for blue light, one cycle
has a length of 460 nm, so the period is 460.
2π
Period = k
2π
460 = k
2π or π
k=
460
230
The equation for blue light waves
x .
is y = sin π
230
In the equation for blue light,
k = π
230
=⋅ 0.014
The approximate equation for blue
light waves is y = sin 0.014x.
5.5 Stretches of Periodic Functions • MHR 373
Key
Concepts
• The vertical and horizontal stretches of sine and cosine functions can be
summarized as follows.
Stretch
Vertical
Mathematical
Form
Effect
If a > 1, the graph is expanded vertically by a factor of a.
y = asin x
y = acos x
If 0 < a < 1, the graph is compressed vertically by a factor of a.
Horizontal y = sin kx
y = cos kx
If k > 1, the graph is compressed horizontally by a factor of 1.
k
If 0 < k < 1, the graph is expanded horizontally by a factor of 1.
k
• For y = asin x or y = acos x, a > 0, the amplitude is a.
2π or 360° .
• For y = sin kx or y = cos kx, k > 0, the period is k
k
Communicate
Yo u r
Understanding
Describe how you would find the amplitude and period of the function
y = 2cos 4x.
y
2. Describe how you would find the amplitude
2
and period of the graph shown.
1.
0
Describe how you would sketch one cycle
of the graph of y = 1 sin 3x, starting
2
at (0, 0), x ≥ 0.
3.
2π
4π
6π
8π x
–2
Describe how you would sketch one cycle of the graph of
y = 5 cos 1x, −π ≤ x ≤ π.
2
4.
Practise
A
Sketch one cycle of the graph of each of
the following. State the domain and range of
the cycle.
a) y = 3sin x
b) y = 5cos x
2
c) y = 1.5sin x
d) y = cos x
3
1.
374 MHR • Chapter 5
Find the period, in degrees and radians,
for each of the following.
a) y = sin 6x
b) y = cos 4x
2
2
c) y = cos x
d) y = 8sin x
3
3
1
e) y = 5sin x
f) y = 7cos 8x
6
2.
Sketch one cycle of the graph of each of
the following. State the domain and range of
the cycle.
a) y = sin 2x
b) y = cos 3x
1
c) y = sin 6x
d) y = cos x
4
3
1
e) y = sin x
f) y = cos x
4
3
c)
y
3.
2
2
a)
y
c)
–2
2
2π
x
0
150° 300° 450° 600°
–4
y
4
0
π
2π
3π
π
–
2
π
3π
––
2
4π x
–2
–4
d)
y
8
4
–4
3π
8π 10π
–2
0
2π
4π 6π
2
1
–4
4
–4
2
–1
y
0
–2
y
–2
b)
2
b)
π
–2
4
Determine the equation for each sine
function.
0
2π x
Determine the equation for each cosine
function.
7.
x
π
8.
Sketch one cycle of the graph of each
of the following.
a) y = 3sin 2x
b) y = 4cos 3x
1
1
5
2
c) y = sin x
d) y = sin x
2
2
3
3
4
e) y = 2sin 4x
f) y = 2.5cos x
5
90° 180° 270° 360°
0
4π x
–4
6.
0
3π
–4
Write the equation and sketch one cycle
for each cosine function described.
a) amplitude 3, period 180°
b) amplitude 0.5, period 720°
c) amplitude 4, period 4π
d) amplitude 2.5, period 5π
2
2π
–2
5.
y
π
0
Write the equation and sketch one
complete cycle for each sine function
described. Each graph begins at the
origin, x ≥ 0.
a) amplitude 6, period 180°
b) amplitude 1.5, period 240°
c) amplitude 0.8, period 3π
d) amplitude 4, period 6π
4
4
4
4.
a)
d)
y
x
x
–8
5.5 Stretches of Periodic Functions • MHR 375
x
9.
a)
b)
c)
d)
Sketch the graphs of the following functions.
y = 4sin x, −2π ≤ x ≤ 2π
y = 1 cos 3x, −π ≤ x ≤ π
2
y = 3cos 4x, −2π ≤ x ≤ π
y = 2sin 1 x, −5π ≤ x ≤ 5π
5
Identify any invariant points when
y = cos x is transformed onto each of the
following.
a) y = 2.5cos x
b) y = cos 3x
c) y = 3cos 2x
d) y = 4cos 0.5x
11.
State the domain, range, amplitude, and
period of each of the following functions.
a) y = 3.5sin x
b) y = cos 2.5x
1
1
1
c) y = 2sin x
d) y = cos x
6
4
2
12.
Identify any invariant points when
y = sin x is transformed onto each of the
following.
a) y = 5sin x
b) y = sin 4x
c) y = 2sin 3x
d) y = 0.5sin 0.5x
10.
Apply, Solve, Communicate
The sine graphs model
waves of yellow, green, and violet light.
The units of the scale on the x-axis are
nanometres. Write equations for the waves
a) using exact values
b) using approximate values, to the
nearest thousandth
13. Colours
y
x = 530
1
x = 410
0
100
200
300
400
500
–1
B
14. Communication Graph y = sin x and
a) For what values of x is sin x > cos x?
b)
c)
y = cos x, for 0 ≤ x ≤ 360°.
For what values of x is sin x < cos x?
For what values of x is sin x = cos x?
15. Sound A pure tone made by a tuning fork can be observed as a sine
curve on a piece of scientific equipment called an oscilloscope. For the
note A, above middle C, the equation of the curve is y = 10sin 880πx.
a) Determine the amplitude and the period.
b) Describe how y = sin x could be transformed onto y = 10sin 880πx.
The frequency of a periodic function is defined as the
number of cycles completed in a second. This is the same as the reciprocal
of the period of a periodic function and is typically measured in Hertz (Hz).
If t is measured in seconds, what is the frequency of
t
a) y = sin t?
b) y = cos t?
c) y = sin 2t?
d) y = 4cos ?
2
16. Application
376 MHR • Chapter 5
600
700
x = 580
800
x
A point on the ocean rises and falls as waves pass.
Suppose that a wave passes every 4 s, and the height of each wave from
the crest to the trough is 0.5 m.
a) Sketch a graph to model the height of the point relative to its average
height for a complete cycle, starting at the crest of a wave.
b) Use exact values to write an equation of the form h(t) = acos kt to model
the height of the point, h(t) metres, relative to its average height, as a
function of time, t seconds.
c) If the times on the t-axis were changed from seconds to minutes, what
would be the transformational effect on the graph, and what would be the
new equation?
17. Ocean cycles
According to biorhythm theory, three cycles affect
people’s lives, giving them favourable and non-favourable days. The physical
cycle has a 23-day period, the emotional cycle has a 28-day period, and the
intellectual cycle has a 33-day period. The cycles can be shown graphically as
sine curves with amplitude 1 and with the person’s date of birth being
considered as the start of each type of cycle.
a) Write a sine function to represent each type of cycle.
b) Graph the cycles for the first 100 days of someone’s life.
c) How old is the person the first time that all three types of cycle reach a
maximum on the same day?
18. Biorhythms
The voltage, V, in volts, of a household alternating current
circuit is given by V(t) = 170sin 120πt, where t is the time in seconds.
a) Determine the amplitude and the period for this function.
b) The number of cycles completed in 1 s is the frequency of the current.
Determine the frequency.
c) Graph two cycles of the function.
19. Electricity
A generator produces a voltage, V, in volts, given by
V(t) = 120cos 30πt, where t is the time in seconds. Graph the function for
0 ≤ t ≤ 0.5.
20. Electricity
C
Use sketches to predict the graphs of
1
1
y = and y = , 0 ≤ x ≤ 4π.
sin x
cos x
b) Use a graphing calculator to check your predictions.
21. Inquiry/Problem Solving a)
5.5 Stretches of Periodic Functions • MHR 377