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