4.6 – Graphs of Composite Trigonometric Functions

Combining the sine function with x2 Graph each of the following functions for Which of the functions appear to be periodic? y = sin x + x2 y = x2 sin x y = (sin x)2 y = sin (x2)

Verifying periodicity algebraically Verify algebraically that the function is periodic and determine its period graphically. f(x) = (sin x)2 f(x) = cos2x f(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:

Adding a sinusoid to a linear function The graph of each function oscillates between what two parallel lines? f(x) = 0.5x + sin x y = 2x + cos x y = 1 – 0.5x + cos 2x

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

Identifying a Sinusoid

You Try! Identifying a Sinusoid

Expressing the sum of sinusoids as a sinusoid Period: Estimate amplitude and phase shift graphically: Give a sinusoid that approximates f(x).

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 bx or cos bx 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

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. Identify the damping factor and tell where the damping occurs Approximately how long does it take for the spring to be damped so that ?

Damped Oscillating Spring