Download WARM UP write each fraction as a decimal and as a percent (a) ¾ (b

WARM UP
Find the amplitude and period of the function and use
the language of transformations to describe how the
graph of the function is related to the graph of y = sin x.
Remember:
Amplitude = |a| Period = 2π/|b|
x
y  3sin
2
3
y   sin 2 x
2
What you’ll learn about
• The Basic Waves Revisited
• Sinusoids and Transformations
• Modeling Periodic Behavior with Sinusoids
… and why
Sine and cosine gain added significance when
used to model waves and periodic behavior.
Sinusoid
Period
Amplitude
Sine and Cosine
f(x) = sin x
•
•
•
•
•
•
f(x) = cos x
Domain: All real numbers
Range: [-1, 1]
Continuous
Alternately increasing and decreasing in periodic waves
Bounded
Absolute maximum of 1; Absolute minimum of 1
Phase Shift
• Remember that the graph of y = f(x +c) is a
translation of the graph of y = f(x) by c units to
the left when c is positive (c is > 0)
• That is exactly what happens with sinusoids,
and we say that the wave undergoes a of
phase shift of –c.
Sin and Cos and Phase Shift
Class Work – Part 1
• Graph f(x) = x2 and f(x) = (x + 1)2. What do you
see?
• Graph f(x) = x3 and f(x) = (x –2)3. What do you
see?
• Graph f(x) = sin x and f(x) = sin (x + 4). What
do you see?
Sin and Cos and Phase Shift
Class Work – Part 2
• Now, graph y = sin(x) in Y1 and y = cos(x) in Y2
• What do you see in terms of a phase shift
between sine and cosine?
• Look at the points where sin(x) has a
maximum and cos(x) has a minimum (and vice
versa)
• Can you write the cosine function as a phase
shift of the sine function, and the sine
function as a phase shift of the cosine
function?
Example Combining a Phase Shift
with a Period Change
Construct a sinusoid with period  /3 and amplitude 4
that goes through (2,0).
Example Combining a Phase Shift
with a Period Change
Construct a sinusoid with period  /3 and amplitude 4
that goes through (2,0).
To find the coefficient of x, set 2 / | b |  / 3 and solve for b.
Find b  6. Arbitrarily choose b  6.
For the amplitude set | a | 4. Arbitrarily choose a  4.
The graph contains (2,0) so shift the function 2 units to the right.
y  4sin(6( x - 2))  4sin(6 x -12).
Frequency of a Sinusoid
• The frequency is simply the reciprocal of the
period.
• Therefore, the frequency of a sinusoid is
|b|/2π
• Graphically, frequency is the number of
complete cycles the wave completes in a unit
interval
Visualizing a Musical Note
• The musical note middle C can be modeled by:
f(x) = 1.5 sin 524πx
• Where x is the time in seconds
• What is the amplitude of this function?
• What is the period of this function?
• What is the frequency of this function? (Frequency is
1/Period)
• Graph the function with the graphing window:
– Xmin = - 60; Xmax = 60
– Ymin = -4; Ymax = 4
– Xscl = π/2
Example: Finding the Frequency of
a Sinusoid
• Find the frequency of the function
f(x) = 4 sin(2x/3) and interpret its meaning
graphically.
• The frequency is 2/3  2π = 1/(3 π)
• This is the reciprocal of the period, which is
3π.
HOMEWORK
P 393 #53 to 56
EXIT TICKET
What is the phase shift between cos x
and sin x?