5.6 Translations and Combinations
of Transformations
The highest tides in the world are found in the
Bay of Fundy. Tides in one area of the bay cause
the water level to rise to 6 m above average sea
level and to drop to 6 m below average sea level.
The tide completes one cycle approximately
every 12 h. The depth of the water can be
modelled by a sine function. This function will
be modelled in Example 6.
I NVESTIGATE & I NQUIRE
Copy and complete the table by finding decimal values for sin x
and sin x + 2. Round values to the nearest tenth, if necessary.
1.
x (degrees)
sin x
sin x + 2
0
0
0
45 90 135 180 225 270 315 360
0.7
2.7
Sketch the graphs of the
functions y = sin x and
y = sin x + 2 on the same
grid, like the one shown.
2.
y
2
0
45
90
135
180
225
270
315
–2
3.
a)
For the graphs from question 2, find
the amplitudes
b) the periods
4. What transformation can been applied to y = sin x to give
y = sin x + 2?
5. a) Make a conjecture about how the graph of y = sin x − 1
compares with the graph of y = sin x.
b) Use your conjecture to sketch the graphs of y = sin x and
y = sin x − 1 on the same grid for the domain 0° ≤ x ≤ 360°.
Test your conjecture from question 5a) by graphing y = sin x and
y = sin x − 1, 0° ≤ x ≤ 360°, on a graphing calculator. Compare the
result with your graph from question 5b).
6.
378 MHR • Chapter 5
360
x
7.
a)
For the graphs from question 6, find
the amplitudes
b) the periods
8.
What transformation can been applied to y = sin x to give y = sin x − 1?
Write a statement about the transformational effect of c on the graph of
y = sin x + c.
9.
Write a conjecture about the transformational effect of c on the graph
of y = cos x + c.
10.
11. Test your conjecture from question 10 by graphing y = cos x,
y = cos x + 2, and y = cos x − 1, 0° ≤ x ≤ 360°, on a graphing calculator.
12. For the graphs
a) the amplitudes
from question 11, find
b) the periods
I NVESTIGATE & I NQUIRE
Copy and complete the tables by finding values of sin x and sin (x + 45°).
Round values to the nearest tenth, if necessary.
1.
x (degrees)
sin x
sin (x + 45)
0 45 90 135 180 225 270 315 360
0 0.7
0.7 1
Sketch the graphs of
the functions y = sin x and
y = sin (x + 45°) on the
same grid, like the one shown.
2.
y
2
0
45
90
135
180
225
270
315
360
x
–2
3.
a)
For the graphs from question 2, find
the amplitudes
b) the periods
4.
What transformation can been applied to y = sin x to give y = sin (x + 45°)?
Make a conjecture about how the graph of y = sin (x − 45°)
compares with the graph of y = sin x.
b) Use your conjecture to sketch the graph of y = sin x and
y = sin (x − 45°) on the same grid for the domain 0° ≤ x ≤ 360°.
5. a)
Test your conjecture from question 5a) by graphing y = sin x and
y = sin (x − 45°), 0° ≤ x ≤ 360°, on a graphing calculator. Compare the result
with your graph from question 5b).
6.
5.6 Translations and Combinations of Transformations • MHR 379
7.
a)
For the graphs from question 6, find
the amplitudes
b) the periods
What transformation can been applied to y = sin x to give
y = sin (x − 45°)?
8.
Write a statement about the transformational effect of d on the graph of
y = sin (x − d).
9.
Write a conjecture about the transformational effect of d on the graph of
y = cos (x − d).
10.
Test your conjecture from question 10 by graphing y = cos x,
y = cos (x + 45°), y = cos (x − 45°), 0° ≤ x ≤ 360°, on a graphing calculator.
11.
12. For the graphs
a) the amplitudes
from question 11, find
b) the periods
The vertical translations that apply to algebraic functions also apply to
trigonometric functions.
If c > 0, the graphs of y = sin x + c and
y = cos x + c are translated upward by
c units.
If c < 0, the graphs of y = sin x + c and
y = cos x + c are translated downward by
c units.
y
y = sin x + 4
4
2
0
y = sin x
45
90
135
–2
–4
As with algebraic transformations, combinations of trigonometric
transformations are performed in the following order.
• expansions and compressions
• reflections
• translations
EXAMPLE 1 Sketching y = asin x + c
Sketch one cycle of the graph of y = 2sin x + 3.
State the domain and range of the cycle.
380 MHR • Chapter 5
180
225
270
y = sin x – 3
315
360
x
SOLUTION
First, sketch the graph of y = 2sin x.
The graph of y = 2sin x is the graph of y = sin x expanded vertically by a
factor of 2.
The amplitude is 2, so the maximum value is 2 and the minimum value
is −2.
The period of the function y = 2sin 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, 0), , 2 , (π, 0), , −2 , and (2π, 0).
2
2
Plot the 5 key points in the cycle. Draw a smooth curve through the points.
Translate the graph three units upward to obtain the graph
of y = 2sin x + 3.
Label the graph.
y
y = 2sin x + 3
4
2
The domain of the cycle is 0 ≤ x ≤ 2π.
The range is 1 ≤ y ≤ 5.
0
–2
π
2π
x
y = 2sin x
The horizontal translations that apply to algebraic functions also apply to
trigonometric functions.
If d > 0, the graphs of y = sin(x − d) and y = cos(x − d) are translated
d units to the right.
If d < 0, the graphs of y = sin(x − d) and y = cos(x − d) are translated
d units to the left.
For trigonometric functions, a horizontal translation is often called the
phase shift or phase angle.
EXAMPLE 2 Sketching y = acos (x – d )
Sketch one cycle of the graph of y = 0.5cos x + π .
2
State the domain, range, and phase shift of the cycle.
5.6 Translations and Combinations of Transformations • MHR 381
SOLUTION
First, sketch the graph of y = 0.5cos x.
The graph of y = 0.5cos x is the graph of y = cos x compressed vertically by
a factor of 0.5.
The amplitude is 0.5, so the maximum value is 0.5 and the minimum value
is −0.5.
The period of the function y = 0.5cos 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, 0.5), , 0 , (π, −0.5), , 0 , and (2π, 0.5).
2
2
Plot the 5 key points in the cycle. Draw a smooth curve through the points.
π
y
Translate the graph units to the left to obtain the graph
2
–
y = 0.5cos x + π
2
1
of y = 0.5cos x + π .
2
0
π
2π x
–π
y = 0.5cos x
Label the graph.
π
3π
The domain of the cycle is – ≤ x ≤ .
2
2
The range is −0.5 ≤ y ≤ 0.5.
π
The phase shift is units to the left.
2
EXAMPLE 3 Sketching y = asin k(x – d )
Sketch one cycle of the graph of y = 3sin 2 x − π .
4
State the domain, range, and phase shift of the cycle.
SOLUTION
First sketch the graph of y = 3sin 2x.
The graph of y = 3sin 2x is the graph of y = sin 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
The period of the function y = 3sin 2x is , or π.
2
382 MHR • Chapter 5
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
3π
(0, 0), , 3 , , 0 , , −3 , and (π, 0).
4
2
4
Plot the 5 key points in the cycle. Draw a smooth curve
through the points.
π
Translate the graph units to the right to obtain the graph
4
π
of y = 3sin 2 x − . Label the graph.
4
5π .
The domain of the cycle is π ≤ x ≤ 4
4
The range is −3 ≤ y ≤ 3.
The phase shift is π units to the right.
4
y
–
y = 3sin 2 x – π
4
2
0
π
2π
x
–2
y = 3sin 2x
If necessary, factor the coefficient of the x-term to identify the characteristics
of a function more easily.
EXAMPLE 4 Sketching y = acos k(x – d ) + c
Sketch the graph of y = 4cos 1x + π − 1, −4π ≤ x ≤ 4π.
2
2
SOLUTION
Factor the coefficient of the x-term.
y = 4cos 1x + π − 1 becomes y = 4cos 1(x + π) − 1.
2
2
2
1
Now, sketch the graph of y = 4cos x.
2
1
The graph of y = 4cos x is the graph of y = cos x expanded vertically by
2
a factor of 4 and expanded horizontally by a factor of 2.
The amplitude is 4, so the maximum value is 4 and the minimum value is −4.
2π , or 4π.
The period of the function y = 4cos 1x is 1
2
2
5.6 Translations and Combinations of Transformations • MHR 383
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
(0, 4), (π, 0), (2π, −4), (3π, 0), and (4π, 4).
Plot the 5 key points in the cycle. Draw a smooth curve through the points.
Translate the graph π units to the left and one unit downward to obtain
the graph of y = 4cos 1(x + π) − 1, −π ≤ x ≤ 3π.
2
Use the pattern to sketch the graph over the domain −4π ≤ x ≤ 4π.
Label the graph.
y
4
1
y = 4cos – x
2
2
–4π
–3π
–2π
–π
π
0
2π
3π
4π x
–2
–4
π
y = 4cos 1
– x + – –1
2
2
EXAMPLE 5 Sketching for a < 1
Sketch the graph of y = −4sin x − π , 0 ≤ x ≤ 4π.
2
SOLUTION
First sketch the graph of y = 4sin x.
The graph of y = 4sin x is the graph of y = sin 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 = 4sin 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, 0), , 4 , (π, 0), , −4 , and (2π, 0).
2
2
Plot the 5 key points in the cycle. Draw a smooth curve through the points.
384 MHR • Chapter 5
y
Recall that the graph of y = −f(x) is the graph of
y = f(x) reflected in the x-axis. So, reflect the graph
of y = 4sin x in the x-axis to obtain the graph of
y = −4sin x.
Translate the reflected graph π units to the right to
2
obtain the graph of
4
y = –4sin x – π–
y = 4sin x
2
2
π
0
3π
2π
4π x
–2
–4
5π .
y = –4sin x
y = −4sin x − π , π ≤ x ≤ 2 2
2
Use the pattern to sketch the graph over the domain 0 ≤ x ≤ 4π. Label the graph.
Note that all of the graphs required in Examples 1−5 can
be drawn directly using a graphing calculator. The graph
shown is y = −4sin x − π , 0 ≤ x ≤ 4π, from Example 5.
2
The calculator is in radian mode, and the window variables include
Xmin = 0, Xmax = 4π, Ymin = −5, and Ymax = 5.
EXAMPLE 6 Bay of Fundy Tides
In one area of the Bay of Fundy, the tides cause the water level to rise to 6 m
above average sea level and to drop to 6 m below average sea level. One cycle is
completed approximately every 12 h. Assume that changes in the depth of the
water over time can be modelled by a sine function.
a) If the water is at average sea level at midnight and the tide is coming in, draw
a graph to show how the depth of the water changes over the next 24 h. Assume
that at low tide the depth of the water is 2 m.
b) Write an equation for the graph.
c) If the water is at average sea level at 02:00, and the tide is coming in, write an
equation for the graph that shows how the depth changes over the next 24 h.
SOLUTION
The depth of water at low tide is 2 m. At low
tide, the water level is 6 m below average sea level.
So, the depth of water for average sea level is 8 m.
This is the depth at midnight. The maximum depth
of the water is 8 + 6, or 14 m. Use the known values
to sketch a 12-h cycle of depth versus time. Use the
pattern to show the changes over 24 h.
d
Water Depth (m)
a)
12
8
4
00:00
04:00
08:00
12:00
16:00
20:00
24:00
Time of Day
5.6 Translations and Combinations of Transformations • MHR 385
The amplitude, a, is 6 m.
The period is 12 h.
2π
12 = k
k = π
6
The graph has been translated 8 units upward, so c = 8.
The equation that shows how the depth of the water changes over time is
πt
h = 6sin + 8.
6
c) When the water is at average sea level at 02:00, the depth is 8 m at 02:00.
The graph is translated 2 h to the right.
The equation is h = 6sin π(t − 2) + 8.
6
b)
Key
Concepts
• Perform combinations of transformations in the following order.
* expansions and compressions
* reflections
* translations
• For trigonometric functions, a horizontal translation is called the phase shift or
phase angle.
• If necessary, factor the coefficient of the x-term to identify the characteristics
of a function more easily.
Communicate
Yo u r
Understanding
1. Describe how you would identify the transformations on y = sin x that result
in each of the following functions.
1
a) y = 3sin x − 3
b) y = 6sin 3(x − 2π) c) y = −2sin (x + π) + 2
2
2. Describe how you would identify the transformations on y = cos x that result
in the function y = 3cos (4x − π).
3. Describe how you would sketch the graph of one cycle of the function
π
y = 2sin 2 x + − 3.
2
386 MHR • Chapter 5
Practise
A
1. Determine the vertical translation and the
phase shift of each function with respect to
y = sin x.
a) y = sin x + 3
b) y = sin x − 1
3π
c) y = sin (x − 45°)
d) y = sin x − 4
e) y = sin (x − 60°) + 1
π
f) y = sin x + + 4
3
3π
g) y = sin x + − 0.5
8
h)
y = sin (x − 15°) − 4.5
Determine the vertical translation and the
phase shift of each function with respect to
y = cos x.
a) y = cos x + 6
b) y = cos x − 3
π
c) y = cos x + d) y = cos (x + 72°)
2
e) y = cos (x − 30°) − 2
π
f) y = cos x + + 1.5
6
g) y = cos (x + 110°) + 25
5π
h) y = cos x − − 3.8
12
2.
3. Sketch one cycle of the graph of each of
the following. State the amplitude, period,
domain, and range of the cycle.
a) y = 3sin x + 2
b) y = 2cos x − 2
1
c) y = 1.5sin x − 1
d) y = cos x + 1
2
Sketch one cycle of the graph of each of
the following. State the amplitude, period,
domain, range, and phase shift of the cycle.
4.
a)
y = cos x − π
2
π
d) y = 3cos x + 4
y = 2sin (x − π)
b)
y = 1 sin x + π
2
2
π
e) y = –cos x + 2
c)
Determine the amplitude, period, vertical
translation, and phase shift for each function
with respect to y = sin x.
a) y = 2sin x − 3
b) y = 0.5sin (2x) − 1
c) y = 6sin 3(x − 20°)
π
d) y = –5sin 2 x − + 1
6
5.
Determine the amplitude, period, vertical
translation, and phase shift for each function
with respect to y = cos x.
a) y = cos x + 3
b) y = cos 3(x − 90°)
π
c) y = –3cos 4 x − + 5
4
2
π
d) y = 0.8cos x − − 7
3
3
6.
Sketch one cycle of the graph of each of
the following. State the amplitude, period,
domain, range, and phase shift of the cycle.
π
a) y = sin 2 x + 4
π
b) y = 2cos 2 x − + 1
4
1
c) y = 3sin (x − π) − 2
2
1
d) y = 4cos (x + 2π) − 4
3
π
e) y = –3sin 2 x − + 2
4
7.
5.6 Translations and Combinations of Transformations • MHR 387
Sketch one cycle of the
graph of each of the following. State the
amplitude, period, domain, range, and phase
shift of the cycle.
π
1
a) y = sin 2x − b) y = cos x − π − 2
2
2
c) y = 2sin (3x − π) + 2
d) y = −3cos (2x − 4π) − 1
8. Communication
9. Write an equation for the function with
the given characteristics, where T is the type,
A is the amplitude, P is the period, V is the
vertical shift, and H is the horizontal shift.
P
V
H
sine
cosine
sine
T
8
7
1
2π
π
4π
–6
2
3
none
none
π right
d) cosine
10
π
2
none
π
left
2
a)
b)
c)
A
Sketch the graph of each of the
following. State the range.
a) y = 2sin x + 2, 0 ≤ x < 3π
b) y = −cos 3x − 2, −π ≤ x ≤ π
π
c) y = 3cos x − , −2π ≤ x ≤ 2π
6
π
d) y = 4sin 2 x + − 1, −π ≤ x ≤ π
4
10.
y = –2sin 2x − π + 1, −π ≤ x ≤ π
3
1
π
f) y = 5cos x − + 2, −π ≤ x ≤ 3π
3
3
π
g) y = 2sin 2x + , −2π ≤ x ≤ 2 π
8
e)
Each graph shows part of the sine
function of the form y = asin k (x − d) + c.
Determine the values of a, k, d, and c for
each graph. Check by graphing.
11.
a)
y
1
π 0
– 2π
— –—
3
3 –1
π
—
3
2π
—
3
x
π
–2
–3
b)
y
6
4
2
–π
0
π
2π
3π
x
–2
Apply, Solve, Communicate
B
The water depth in a harbour is 21 m at high tide and
11 m at low tide. One cycle is completed approximately every 12 h.
a) Find an equation for the water depth as a function of the time,
t hours, after low tide.
b) Draw a graph of the function for 48 h after low tide, which occurred
at 14:00.
12. Ocean cycles
388 MHR • Chapter 5
c)
i)
d)
i)
e)
i)
State the times at which the water depth was
a maximum
ii) a minimum
iii) at its average value
Estimate the depth of the water at
17:00
ii) 21:00
Estimate the times at which the depth of the water was
14 m
ii) 20 m
iii) at least 18 m
An object attached to the end of a spring is oscillating up
and down. The displacement of the object, y, in centimetres, is a function
of the time, t, in seconds, and is given by y = 2.4cos 12t + π .
6
a) Sketch two cycles of the function.
b) What is the maximum distance through which the object oscillates?
c) What is the period of the function? Give your answer as an exact
number of seconds, in terms of π, and as an approximate number of
seconds, to the nearest hundredth.
13. Application
On a certain day, the depth of water off a pier at high
tide was 6 m. After 6 h, the depth of the water was 3 m. Assume a 12-h
cycle.
a) Find an equation for the depth of water, with respect to its average
depth, in terms of the time, t hours, since high tide.
b) Draw a graph of the depth of water versus time for 48 h after high tide.
c) Find the depth of water at t = 8 h, 15 h, 20 h, and 30 h.
d) Predict how the equation will change if the first period begins at low
tide.
e) Test your prediction from part d) by drawing the graph and finding the
equation.
14. Ocean cycles
An object suspended from a spring is oscillating up and
down. The distance from the high point to the low point is 30 cm, and the
object takes 4 s to complete 5 cycles. For the first few cycles, the distance
from the mean position, d(t) centimetres, with respect to the time, t
seconds, is modelled by a sine function.
a) Sketch a graph of this function for two cycles.
b) Write an equation that describes the distance of the object from its
mean position as a function of time.
15. Spring
5.6 Translations and Combinations of Transformations • MHR 389
A carnival Ferris wheel with a radius of 7 m makes one
complete revolution every 16 s. The bottom of the wheel is 1.5 m above the
ground.
a) Draw a graph to show how a person’s height above the ground varies
with time for three revolutions, starting when the person gets onto the Ferris
wheel at its lowest point.
b) Find an equation for the graph.
c) Predict how the graph and the equation will change if the Ferris wheel
turns more slowly.
d) Test your predictions from part c) by drawing the graph for three
revolutions and finding an equation, if the wheel completes one revolution
every 20 s.
16. Ferris wheel
A Ferris wheel has a radius of 10 m and makes
one revolution every 12 s. Draw a graph and find an equation to
show a person’s height above or below the centre of rotation for
two counterclockwise revolutions starting at
a) point A
b) point B
c) point C
17. Ferris wheel
For the sine function expressed in the form y = asin k (x − d) + c, does
the value of c affect each of the following? Explain.
a) the amplitude?
b) the period?
c) the maximum and minimum values of the function?
d) the phase shift?
18.
The depth of water, d(t) metres, in a seaport can be
approximated by the sine function d(t) = 2.5sin 0.164π(t − 1.5) + 13.4,
where t is the time in hours.
b) Graph the function for 0 ≤ t ≤ 24 using a graphing calculator.
c) Find the period, to the nearest tenth of an hour.
d) A cruise ship needs a depth of at least 12 m of water to dock safely. For
how many hours in each period can the ship dock safely? Round your
answer to the nearest tenth of an hour.
19. Ocean cycles
390 MHR • Chapter 5
C
B
30°
30°
30°
A
Pose a problem related to each of the
following. Check that you are able to solve each problem. Then, have a
classmate solve it.
a) the vertical motion of a spring
b) the motion of a wheel
20. Pose and solve problems
C
The equation of a sine function can be expressed in the form
y = asin k (x − d) + c.
Describe what you know about a, k, d, and c for each of the following
statements to be true.
a) The period is greater than 2π.
b) The amplitude is less than one unit.
c) The graph passes through the origin.
d) The graph has no x-intercepts.
21.
Predict how the graphs of y = sin (x + 45°) and y = sin(x − 315°)
are related.
b) Test your prediction using a graphing calculator, and explain your
observations.
22. a)
Predict how the graphs of y = sin x and y = cos x − π are related.
2
b) Test your prediction using a graphing calculator, and explain your
observations.
23. a)
A C H I E V E M E N T Check
Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application
The rodent population in a particular region varies
We b C o n n e c t i o n
with the number of predators that inhabit the
www.school.mcgrawhill.ca/resources/
region. At any time, you could predict the
To investigate a simulation of a predator-prey
rodent population, r(t), using the function
relationship, visit the above web site. Go to
r(t) = 2500 + 1500sin πt , where t is
Math Resources, then to MATHEMATICS 11,
4
the number of years that have passed since
to find out where to go next. Report on your
1976.
findings.
a) In the first cycle of this function, what was the
maximum number of rodents and in which year did this
occur?
b) What was the minimum number of rodents in a cycle?
c) What is the period of this function?
d) How many rodents would you predict in the year 2014?
e) Change the function to model a rodent population cycle that lasts 5 years.
5.6 Translations and Combinations of Transformations • MHR 391