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4.6 – Graphs of Composite Trigonometric Functions
Combining the sine function with x2
a) y = sin x + x2
a) y = x2 sin x
a) y = (sin x)2
a) y = sin (x2)
Graph each of the following functions for Which of the functions appear to be periodic?
−2π ≤ x ≤ 2π
Verifying periodicity algebraically
f(x) = (sin x)2
f(x) = cos2x
f(x) =
Verify algebraically that the function is periodic and determine its period graphically.
cos2 x
Composing y = sin x and y = x3
Prove algebraically that f(x) = sin3x is periodic and find the period graphically:
Analyzing nonnegative periodic functions
Domain: Range: Period:
Domain: Range: Period:
g(x)=sinx
g(x)=cotx
Adding a sinusoid to a linear function
f(x) = 0.5x + sin x
y = 2x + cos x
y = 1 – 0.5x + cos 2x
The graph of each function oscillates between what two parallel lines?
Sums that are Sinusoid Functions
If y1 = a1sin(b(x-h1))
and y2 = a2 cos (b(x-h2)) then
y1 + y2 = a1 sin (b(x-h1)) + a2 cos (b(x-h2))
is a sinusoid with period
2πb
Identifying a Sinusoid
f (x)=sinx−3cosx
f (x)=2cosπx+sinπx
f (x)=3sin2x−5cosx
You Try! Identifying a Sinusoid
f (x)=2cos3x−3cos2x
f (x)=5cosx+ 3sinx
f (x)=acos3x7
⎛⎝⎜
⎞⎠⎟−bcos
3x7
⎛⎝⎜
⎞⎠⎟+csin
3x7
⎛⎝⎜
⎞⎠⎟
Expressing the sum of sinusoids as a sinusoid
Period:
Estimate amplitude and phase shift graphically:
Give a sinusoid that approximates f(x).
asin(b(x−h))
f (x)=2sinx+5cosx
Showing a function is periodic but not a sinusoid
f(x) = sin 2x + cos 3x
f(x) = 2 cos x + cos 3x
Damped Oscillation
What happens when sin bt or cos bt is multiplied by another function.
Ex: y = (x2 + 5) cos 6x
Damped Oscillation
The graph of y = f(x) cos bx or y = f(x) sin bx
oscillates between the graphs of y = f(x) and y = -f(x).
When this reduces the amplitude of the wave, it is called damped oscillation. The factor of f(x) is called the damping factor.
Identifying a damped oscillation
f (x)=2−xsin4x
f (x)=3cos2x
f (x)=−2xcos2x
A damped oscillation spring
Ms. Samara’s Precalculus class collected data for an air table glider that oscillated between two springs. The class determined from the data that the equation :
Modeled the displacement y of the spring from its original position as a function of time t.
y=0.22e−0.065t cos2.4t
a) Identify the damping factor and tell where the damping occurs
b) Approximately how long does it take for the spring to be damped so that ? −0.1≤ y ≤ 0.1
Damped Oscillating Spring
Homework
Pg. 413-414:2, 8, 12, 18, 22, 26, 34, 36, 39-42, 44, 45, 52, 56, 62, 66, 70