The document defines sinusoidal functions as functions that can be written in the form f(x) = asin(bx - c) + d or f(x) = acos(bx - c) + d, where a, b, c, and d are real numbers. It lists some key properties of sinusoidal functions, including their period (2π/b), amplitude (a), and phase shift (-c/b). It provides examples of sketching sinusoidal functions and determining their properties based on given equations or graphs.

1. Sinusoidal Functions

2. Definition:
A function that can be written in the form:
f(x) = asin(bx ­ c) + d
or
f(x) = acos(bx ­ c) + d
Where a, b, c, and d are real numbers

3. Property
2
Period
b
Amplitude a
­c
Phase Shift
b
Maximum d + a
Minimum d ­ a
Range [ d ­ a , d + a
]

4. Sketch y = sin4x Determine the period

5. Determine the period
Graph y = cos(x ­ )
2

6. Graph f(x) = 3sin(2x ­ ) + 1

7. Given this graph determine 2 possible equations

11. At a sea port, the depth of the water, h meters, at time, t hours, during a certain
day is given by this formula:
(a) State the: (i) period (ii) amplitude (iii) phase shift.

12. At a sea port, the depth of the water, h meters, at time, t hours, during a certain
day is given by this formula:
(b) What is the maximum depth of the water? When does it occur?

13. At a sea port, the depth of the water, h meters, at time, t hours, during a certain
day is given by this formula:
(c) Determine the depth of the water at 5:00 am and at 12:00 noon.
(d) Determine one time when the water is 2.25 meters deep.

14. At a sea port, the depth of the water, h meters, at time, t hours, during a certain
day is given by this formula:
(d) Determine one time when the water is 2.25 meters deep.

15. Exercise 13, questions 1 ­ 9