14.5 Write Trigonometric Functions and Models
Goal · Model data using sine or cosine functions.
Your Notes
VOCABULARY
Sinusoids
Graphs of sine and cosine function
EXAMPLE 1
Solve a multi-step problem
Write a function for the
sinusoid shown below.
Solution
Step 1 The maximum value M of the function is _5_
and the minimum value m of the function is _-3_ .
Step 2 The value of k is the mean of the maximum and
minimum values. The vertical shift is
Step 3 When x = 0, the function is at its minimum. Use a _cosine_
function whose graph is a reflection in the x-axis with no horizontal
shift. So, h = _0_ .
Step
4 The
period is _4p_ = So, b = The
amplitude is
Because the graph is a reflection, a
_<_ 0. So, a =
_-4_ .
The function is y
Your Notes
Example 2
Use sinusoidal regression
Temperature The table below shows the average monthly high temperature H
(in °F) for
|
t |
1 |
2 |
3 |
4 |
5 |
6 |
|
|
|
|
|
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|
H |
32 |
34 |
43 |
55 |
65 |
75 |
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|
|
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|
t |
7 |
8 |
9 |
10 |
11 |
12 |
|
||||||||||
|
H |
81 |
79 |
73 |
61 |
47 |
36 |
|
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Solution
Begin by entering the data in a
graphing calculator. Use the graphing calculator's sinusoidal regression
feature to get a model.
H= __24.7 sin(0.513t- 2.15) + 56.3__
Checkpoint Complete the following exercises.
- Write a function for the sinusoid.
y = 6 sin(p x) + 1
- The
table shows the average monthly low temperature L (in °F) for
|
t |
1 |
2 |
3 |
4 |
5 |
6 |
|
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|
L |
18 |
20 |
29 |
40 |
50 |
60 |
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|
|
|
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|
t |
7 |
8 |
9 |
10 |
11 |
12 |
|
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|
L |
66 |
65 |
58 |
47 |
34 |
23 |
|
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L = 24.1 sin(0.510t- 2.15) + 41.9