1. 4.6 – Graphs of Composite Trigonometric Functions

2. 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)

3. 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) =

4. Composing y = sin x and y = x3
Prove algebraically that f(x) = sin3x is periodic and find the period graphically:

5. Analyzing nonnegative periodic functions
Domain:
Range:
Period:

6. 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

7. 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

8. Identifying a Sinusoid

9. You Try! Identifying a Sinusoid

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

11. Showing a function is periodic but not a sinusoid
f(x) = sin 2x + cos 3x
f(x) = 2 cos x + cos 3x

12. Damped Oscillation
What happens when sin bx or cos bx is multiplied by another function? Ex: y = (x2 + 5) cos 6x

13. 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.

14. Identifying a damped oscillation

15. 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 ?

16. Damped Oscillating Spring