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LESSON
4:TRANSFORMATIONS HOMEWORK QUESTIONS
Quick Review:
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x (radians) |
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x (degrees) |
0 |
30 |
60 |
90 |
120 |
150 |
180 |
210 |
240 |
270 |
300 |
330 |
360 |
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sin x (exact) |
0 |
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1 |
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-1 |
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sin x (approx.) |
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0.5 |
0.87 |
1 |
0.87 |
0.5 |
0 |
-0.5 |
-0.87 |
-1 |
-0.87 |
-0.5 |
0 |
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x (radians) |
0 |
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x (degrees) |
0 |
30 |
60 |
90 |
120 |
150 |
180 |
210 |
240 |
270 |
300 |
330 |
360 |
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cos x (exact) |
1 |
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-1 |
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1 |
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cos x (approx.) |
1 |
0.5 |
0.87 |
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-0.5 |
-0.87 |
-1 |
-0.87 |
-0.5 |
0 |
0.5 |
0.87 |
1 |
![Text Box: In summary, to graph y = a sin [k(x – d)] + c from the graph of y = sin(x), follow these ideas:
· If a < 0, we have a reflection in the x-axis
· If k < 0, we have a reflection in the y-axis
· If | a | < 1, we have a vertical compression , factor | a |
· If | a | > 1, we have a vertical stretch, factor | a |
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· If | k | < 1, we have a horizontal stretch, factor 1/k
· If | k | > 1, we have a horizontal compression, factor 1/k
· The value of d gives the horizontal translation (phase shift)
· The value of c gives the vertical translation (shift)](./transhw2_files/image072.gif)
Homework
questions cont’d: (Solutions below)
1. State the amplitude, period, phase shift,
domain and range for each of the following trigonometric functions. Draw the graph of at least
one
complete period for each.
2. The graph below shows a sine function of the form y = a sin k(x – d) + c. find the values of the parameters a,k, d, c.
3. The graph below shows a sine function of the
form y = a cos k(x
– d) + c. find the values
of the parameters a,k, d, c.
Solutions:
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x0 |
0 |
90 |
180 |
270 |
360 |
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y |
0 |
1 |
0 |
-1 |
0 |
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x0 |
0 |
90 |
180 |
270 |
360 |
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y |
0 |
1 |
0 |
-1 |
0 |
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x0 |
0 |
90 |
180 |
270 |
360 |
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y |
1 |
0 |
-1 |
0 |
1 |
In radians,
the points (see table at top of page) would be:
Amplitude: |a| = 2
Domain is
given as:
Range: The maximum
value of y is 3 and the minimum value is –1
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x0 |
0 |
90 |
180 |
270 |
360 |
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y |
0 |
1 |
0 |
-1 |
0 |
(x, y) -----------------------à (½ x + 22.50, 3y + 1)
Using the
points from the table above we get points for the transformed graph
(x, y) -----------------------------à (½ x + 22.50, 3y +1)
(0, 0) ------------------------à (22.50,
1)
(900,
1) ----------------------à (67.50, 4)
(1800,
0) ---------------------à (112.50, 1)
(2700
, -1) --------------------à (157.50, -2)
(3600,
0) ----------------------à (202.50, 1)
[red graph at
left]
In radians, the points (see table
at top of page) would be:
Amplitude: |a| =
3
Domain is given as:
Range: The maximum value of y is 4 and the
minimum value is – 2
2. The graph below shows a sine function of the form y = a sin k(x – d) + c. find the values of the parameters a,k, d, c.
Solution:
Recall
the basic sine curve starts at (0, 0).
If this point is shifted right 450 and down 1, we end up on
the given curve.
Hence d
= 450 and c = -1, yielding the equation:
3. The graph below shows a sine function of the
form y = a cos k(x
– d) + c. find the values
of the parameters a,k, d, c.
Solution:
Recall
the basic cosine curve starts at (0, 1).
If this point is shifted right 300 and up 0.5, we end up on
the given curve.
Hence d
= 300 and c = 0.5, yielding the equation:
