The general form of the sine function and the various transformations that may be done to it: amplitude, period, phase shift, vertical shit and reflections.

1. Let's look at the weather ...
Winnipeg Weather Data as of May 15, 2007 for the last year
Temperature
Month J F MAM J J A SO N D
Mean -17 -14 -6 4 12 17 20 18 12 6 -4 -14
Source: Winnipeg weather statistics

3. Hours of Sunshine
Month J F MAM J J A SON D
Mean 120 140 178 232 277 291 322 286 189 150 95 99
Source: Winnipeg weather statistics
swivel your data

4. Properties and Transformations of the sine function ...
Let's look at some graphs ...

5. In general form, the equation and graph of the basic sine function is:
ƒ(x) = AsinB(x - C) + D
A=1, B=1, C=0, D=0
2π
-2π
-π π
Note that your calculator displays:
ƒ(x) = asin(bx - c) + d
The quot;starting point.quot;
Which is equivalent to:
ƒ(x) = AsinB(x - c/b) + D
In general form, the equation and graph of the basic cosine function is:
ƒ(x) = AcosB(x - C) + D The quot;starting point.quot;
-2π 2π
Since these graphs are so similar
(they differ only by a quot;phase -π π
shiftquot; of π/2 units) we will limit A=1, B=1, C=0, D=0
our study to the sine function.

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.

7. The Role of Parameter B
B is not the period; it determines the period according to this relation:
or

8. The Role of Parameter C
C is called the phase shift, or horizontal shift, of the graph.

9. 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.