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Lesson 6.6 Exercises, pages 534–539
A
3. Identify the transformations that would be applied to the graph of
1
y ⫽ sin x to get the graph of y ⫽ 10 sin 3 (x ⫺ ␲) ⫹ 1.
1
3
Compare y ⴝ 10 sin (x ⴚ ␲) ⴙ 1 with y ⴝ a sin b(x ⴚ c) ⴙ d:
a ⴝ 10, so the graph of y ⴝ sin x is stretched vertically by a factor of 10.
1
3
b ⴝ , so the graph of y ⴝ sin x is stretched horizontally by a factor of 3.
c ⴝ ␲, so the graph of y ⴝ sin x is translated ␲ units right.
d ⴝ 1, so the graph of y ⴝ sin x is translated 1 unit up.
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6.6 Combining Transformations of Sinusoidal Functions—Solutions
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4. Identify the following characteristics of the graph below: amplitude;
period; phase shift; equation of the centre line; zeros; domain;
maximum value; minimum value; range
2
y 2 sin
1
x
2
6
(
)
y
1
x
4 2 3
3
3
0
1
3
2
3
4
3
5
3
2
The amplitude is 2. The period is 4␲. The phase shift is
of the centre line is y ⴝ 0. The zeros are ⴚ
␲
. The equation
6
␲
11␲
and . The graph is
6
6
shown on domain ⴚ2␲ ◊ x ◊ 2␲. The maximum value is 2. The
minimum value is ⴚ2. The range is ⴚ2 ◊ y ◊ 2.
B
5. Use the given data to write an equation for each function.
a) a sine function with: amplitude 5; period 3␲; equation of centre
␲
line y ⫽ ⫺2; and phase shift 3
Use: y ⴝ a sin b(x ⴚ c) ⴙ d
Since the period ⴝ
2
2␲
2␲
, then b ⴝ , or
3␲
3
b
2
3
In y ⴝ a sin b(x ⴚ c) ⴙ d, substitute: a ⴝ 5, b ⴝ , c ⴝ
␲
3
An equation is: y ⴝ 5 sin ax ⴚ b ⴚ 2
2
3
␲
, d ⴝ ⴚ2
3
b) a cosine function with: maximum value 5; minimum value ⫺2;
␲
period ␲; and phase shift ⫺ 4
Use: y ⴝ a cos b(x ⴚ c) ⴙ d
5 ⴚ ( ⴚ 2)
From the maximum and minimum values, a ⴝ
, or 3.5
2
2␲
From the period, b ⴝ ␲ , or 2
From the maximum value and the amplitude, d ⴝ 5 ⴚ 3.5, or 1.5
␲
In y ⴝ a cos b(x ⴚ c) ⴙ d, substitute: a ⴝ 3.5, b ⴝ 2, c ⴝ ⴚ , d ⴝ 1.5
4
␲
4
An equation is: y ⴝ 3.5 cos 2 ax ⴙ b ⴙ 1.5
6. Determine a possible equation for each function graphed below.
a)
1
0
1
y
y f(x)
3
©P DO NOT COPY.
2
3
x
2
Sample response: The graph is
the image of y ⴝ sin x after a
vertical compression by a
1
2
factor of .
An equation is: y ⴝ
1
sin x
2
6.6 Combining Transformations of Sinusoidal Functions—Solutions
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b)
1
Sample response:
The graph is the image of
y ⴝ cos x after a vertical
translation of 2 units down.
An equation is: y ⴝ cos x ⴚ 2
y
3
0
1
2
3
4
3
5
3
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x
2
2
y f (x)
3
c)
1
Sample response:
The graph is the image of
y ⴝ cos x after a horizontal
y f(x)
y
x
2
0
1
4
compression by a factor of .
1
d)
1
An equation is: y ⴝ cos 4x
Sample response:
The graph is the image of
y ⴝ cos x after a horizontal
␲
translation of units right.
y
y f(x)
3
0
1
x
2
3
␲
3
An equation is: y ⴝ cos ax ⴚ b
7. a) For the function graphed below, identify the values of a, b, c, and
d in y ⫽ a sin b(x ⫺ c) ⫹ d, then write an equation for the
function.
6
y
4
2
4 2 3
3
3
0
y f(x)
3
2
3
x
4
3
Sample response: The equation of the centre line is y ⴝ 4, so the
vertical translation is 4 units up and d ⴝ 4.
The amplitude is:
6ⴚ2
ⴝ 2, so a ⴝ 2
2
Choose the x-coordinates of two adjacent maximum points, such as
and
2␲
5␲
5␲
␲
. The period is:
ⴚ ⴝ
6
6
6
3
So, b is:
␲
6
2␲
ⴝ3
2␲
3
The sine function begins its cycle at x ⴝ 0; so the phase shift is 0, and
c ⴝ 0.
Substitute for a, b, c, and d in: y ⴝ a sin b(x ⴚ c) ⴙ d
An equation is: y ⴝ 2 sin 3x ⴙ 4
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b) For the function graphed below, identify the values of a, b, c, and
d in y ⫽ a cos b(x ⫺ c) ⫹ d, then write an equation for the
function.
y
1
y f(x)
x
3 4
2
4
4
0
1
2
3
4
Sample response: The equation of the centre line is y ⴝ ⴚ1, so the
vertical translation is 1 unit down and d ⴝ ⴚ1.
The amplitude is:
ⴚ 0.5 ⴚ ( ⴚ 1.5)
1
ⴝ 0.5, so a ⴝ
2
2
Choose the x-coordinates of two adjacent maximum points, such as
and
␲
5␲
5␲
␲
. The period is:
ⴚ ⴝ
8
8
8
2
␲
8
2␲
ⴝ4
␲
2
␲
To the right of the y-axis, the cosine function begins its cycle at x ⴝ ,
So, b is:
␲
8
so the phase shift is , and c ⴝ
8
␲
.
8
Substitute for a, b, c, and d in: y ⴝ a cos b(x ⴚ c) ⴙ d
An equation is: y ⴝ
␲
1
cos 4 ax ⴚ b ⴚ 1
2
8
8. a) The graph of y ⫽ sin x is shown below. On the same grid, sketch
␲
the graph of y ⫽ 2 sin 3 ax - 2 b ⫹ 3. Describe your strategy.
5
y
(
y 2 sin 3 x 3
2
)
4
3
2
1
x
2
0
1
2
3
2
y sin x
The graph of y ⴝ sin x is: stretched vertically by a factor of 2,
1
3
compressed horizontally by a factor of , then translated
␲
units
2
right and 3 units up
I chose points on the graph of y ⴝ sin x, applied the transformations to
each point, then joined the image points.
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␲
b) List the characteristics of the function y ⫽ 2 sin 3 ax - 2 b ⫹ 3.
The amplitude is 2; the period is
is x ç ⺢; the range is 1
␲
2␲
; the phase shift is ; the domain
3
2
◊ y ◊ 5; there are no zeros.
9. a) The graph of y ⫽ cos x is shown below. On the same grid, sketch
␲
the graph of y ⫽ cos 4 ax + 3 b ⫺ 2. Describe your strategy.
2
y
y cos 4x
1
y cos x
x
2
0
1
2
(
y cos 4 x 2
3
)
1
4
The graph of y ⴝ cos x is: compressed horizontally by a factor of , then
translated
␲
units left and 2 units down.
3
I first graphed y ⴝ cos 4x, then chose points on this graph and applied
the remaining transformations to each point. I continued the pattern of
image points, then joined them.
␲
b) List the characteristics of the function y ⫽ cos 4 ax + 3 b ⫺ 2.
The amplitude is 1; the period is
␲
2␲
␲
ⴝ ; the phase shift is ⴚ ;
4
2
3
the domain is x ç ⺢; the range is ⴚ3
◊ y ◊ ⴚ1; there are no zeros.
10. Sketch the graph of each function for the domain ⫺2␲ ≤ x ≤ 2␲.
␲
1
a) y ⫽ 4 sin 2 ax - 3 b ⫺ 3
4
1
y 4 sin x
2
y
2
x
5 4 2
3
3
3
3
3
0
2
y 4 sin
4
4
3
5
3
1
x
3
2
3
(
)
6
1
␲
Sketch the graph of y ⴝ 4 sin x, then translate it units right and
2
3
3 units down.
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␲
b) y ⫽ 3 cos 2 ax + 4 b ⫹ 1
1
y
1
cos 2 x 1
3
4
(
y
)
1
x
0
1
y cos 2x
3
␲
1
Sketch the graph of y ⴝ cos 2x, then translate it units left and
3
4
1 unit up.
C
␲
11. Use transformations to sketch the graph of y ⫽ ⫺2 sin a2x + 3 b ⫺ 2
for ⫺2␲ ≤ x ≤ 2␲.
y
y 2 sin 2x
y 2 sin 2x
2
x
0
4
6
(
y 2 sin 2x 2
3
)
␲
Write the function as y ⴝ ⴚ2 sin 2 ax ⴙ b ⴚ 2.
6
Sketch the graph of y ⴝ 2 sin 2x, reflect it in the x-axis to get the graph of
y ⴝ ⴚ2 sin 2x, then translate this graph
©P DO NOT COPY.
␲
units left and 2 units down.
6
6.6 Combining Transformations of Sinusoidal Functions—Solutions
35