This document discusses sinusoidal functions and their properties and transformations. It explains the role of each parameter (A, B, C, D) in the general sinusoidal equation A sin B(x - C) + D. It provides examples of how changing each parameter affects the graph of the sine function. It also includes homework questions asking students to determine parameter values and properties from graphs.

1. Learning to
Read the Wave
Riding the Perfect Wave by
flickr user San Diego Shooter

2. Generate a sinusoidal equation and sketch the graph for the following
set of data. Write the values of the parameters in your equation to two
decimal places.
Homework
Amount of Precipitation in Winnipeg (in mm)
Month J F M A M J J A S O N D
mm 19 15 23 36 60 84 72 75 51 30 21 19
Source: Winnipeg weather statistics

3. Generate a sinusoidal equation and sketch the graph for the following
set of data. Write the values of the parameters in your equation to two
decimal places.
Homework
The data below show the population of a species of marmot in a
given area over a 9-year period on June 1st of each year.
Year 0 1 2 3 4 5 6 7 8
Population 200 188 160 132 120 133 161 187 201

4. Properties and Transformations of the sine function ...
Let's look at some graphs ...
http://fooplot.com
ƒ(x) = AsinB(x - C) + D

5. The Role of Parameter D
D is the sinusoidal axis, average value of the function, or the
vertical shift.
D > 0 the graph shifts up D units. D < 0 the graph shifts down D units.

6. The Role of Parameter A
The amplitude is the absolute value of A; |A|. It is the distance
from the sinusoidal axis to a maximum (or minimum). If it is
negative, the graph is reflected (flips) over the sinusoidal axis.
y = 2sin(x)
y = 1 sin(x)
2
y = -3sin(x)

7. Properties and Transformations of the sine function ...
Let's look at some graphs ...
http://fooplot.com
ƒ(x) = AsinB(x - C) + D

8. y = 2sin(x) + 3

9. The Role of Parameter B
B is not the period; it determines the period according to this relation:
or
y = sin(2x)
y = sin(3x)

10. The Role of Parameter C
C is called the phase shift, or horizontal shift, of the graph.
y = sin(x + π )
4
y = sin(x - π )
4

11. In general form, the equation and graph of the basic sine function is:
ƒ(x) = AsinB(x - C) + D
-2π -π π 2π
A=1, B=1, C=0, D=0
The quot;starting point.quot;
Note that your calculator displays:
ƒ(x) = asin(bx - c) + d
Which is equivalent to:
ƒ(x) = AsinB(x - c/b) + D

12. How many periods are illustrated in each graph? HOMEWORK
How many revolutions (in radians and degrees) are illustrated in
each graph?
Periods =
Radians Rotated =
Degrees Rotated =
Periods =
Radians Rotated =
Degrees Rotated =
Periods =
Radians Rotated =
Degrees Rotated =

13. Determine approximate values for the parameters 'a', 'b', 'c', and 'd'
from the graphs, and then write the equations of each graph as a
sinusoidal function in the form: y = a sin b(x - c) + d HOMEWORK
Re
quot;D me
AB mb
C! er
quot;

14. State the amplitude, period, horizontal shift, and vertical shift for each
of the following: HOMEWORK
amplitude:
period:
horizontal shift:
vertical shift:
amplitude:
period:
horizontal shift:
vertical shift: