The document provides instructions on graphing sine and cosine functions by identifying key points, describing how amplitude, period, and phase shift affect the graph, and giving examples of constructing sinusoidal models from data. It explains how to find the amplitude, period, phase shift, and vertical shift from maximum and minimum values of a periodic function and choose between sine and cosine based on the behavior at a given time. Examples demonstrate modeling tide depth from high and low tide times and depths.

1. LESSON
4.4 Graphs of Sine and Cosine
Quick Review
1st find key points of the graph (where the values are 1, 0, -1) and make a table
Ө sinӨ 2nd we remember what every variable does to the
0 0 original sinusoidal form.
π/2 1
π 0
Shifts vertically
3π/2 -1
+ up, - down
2π 0 (We will learn of this today)
-h shifts right, +h shifts left (we will learn of this Today)
Stretches or shrinks horizontally
Amplitude=shrinks or stretches vertically

2. LESSON
4.4 Graphs of Sine and Cosine
Quick Review
3rd the -3 in amplitude is a vertical stretch and a reflection across the x-axis, o all
we do is multiply our y-values by -3
Ө(x-values) sinӨ (y-values)
Ө 3
00 0 -3π
ππ/2 -3
2ππ 0
3π3π/2 3 -3
3π
4π2π 0
5th now we graph

3. LESSON
4.4 Graphs of Sine and Cosine
Graphs of Sinusoids
The graphs of y  a sin(b( x  h))  k and y  a cos(b( x  h))  k
(where a  0 and b  0) have the following characteristics:
amplitude = |a | ;
2
period = ;
|b|
|b|
frequency = .
2
When complared to the graphs of y  a sin bx and y  a cos bx,
respectively, they also have the following characteristics:
a phase shift of h;
a vertical translation of k .
To the web

4. LESSON
4.4 Graphs of Sine and Cosine
Example Combining a Phase Shift with a
Period Change
Construct a sinusoid with period  /3 and amplitude 4
that goes through (2,0).
A= 4

5. LESSON
4.4 Graphs of Sine and Cosine
Example Give the equation for the Graph
1. Recognize type of function
2. Determine known values
3. Solve
(2π, -3)

6. LESSON
4.4 Graphs of Sine and Cosine
Constructing a Sinusoidal Model using Time
1. Determine the maximum value M and minimum value m.
M m
The amplitude A of the sinusoid will be A  , and
2
M m
the vertical shift will be C  .
2
2. Determine the period p, the time interval of a single cycle
of the periodic function. The horizontal shrink (or stretch)
2
will be B  .
p
Slide 4- 6

7. LESSON
4.4 Graphs of Sine and Cosine

8. LESSON
4.4 Graphs of Sine and Cosine
Constructing a Sinusoidal Model using Time
3. Choose an appropriate sinusoid based on behavior
at some given time T . For example, at time T :
f (t )  A cos( B(t  T ))  C attains a maximum value;
f (t )   A cos( B(t  T ))  C attains a minimum value;
f (t )  A sin( B(t  T ))  C is halfway between a minimum
and a maximum value;
f (t )   A sin( B (t  T ))  C is halfway between a maximum
and a minimum value.

9. LESSON
4.4 Graphs of Sine and Cosine
Pedro records the water level in Lacharca Bay every 2 hours
starting at midnight. Plot the data and find the equation that
models it.
T (hours) 0 2 4 6 8 10 12 14 16 18 20 22 24
Depth (m) 8 10 8 4 2 4 8 10 8 4 2 4 8

10. LESSON
4.4 Graphs of Sine and Cosine
Example Constructing a Sinusoidal Model
On a certain day, high tide occurs at 7:12 AM and the
water depth is measured at 15 ft. On the same day, low
tide occurs at 1:24 and the water depth measures 8 ft.
(a) Write a sinusoidal function modeling the tide.
(b) What is the approximate depth of water at 11:00 AM?
At 3:00 PM?

11. LESSON
4.4 Graphs of Sine and Cosine
Example Constructing a
Sinusoidal Model
(c) At what time did the first low tide occur? The second
high tide?

12. LESSON
4.4 Graphs of Sine and Cosine
Homework exercises 35---77 (every other odd)
Bonus74 and 76
Pages 392---394