More on transformations of periodic functions (the sine function).

1. Applications of
Periodic Functions
or
Bugs On Wheels
Suicidal Shield Bug by flickr user ChinchillaVilla

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

3. The Role of Parameter D ƒ(x) = AsinB(x - C) + 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.

5. The Role of Parameter A ƒ(x) = AsinB(x - C) + D
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.

6. The Role of Parameter B ƒ(x) = AsinB(x - C) + D
B is not the period; it determines the period according to this relation:
or

7. The Role of Parameter C ƒ(x) = AsinB(x - C) + D
C is called the phase shift, or horizontal shift, of the graph.
WATCH THE SIGN OF C
when C < 0 the when C > 0 the
graph shifts left graph shifts right
ƒ(x) = AsinB(x - C) + D
ƒ(x) = asin(bx - c) + d
c = BC

8. 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
Note that your calculator displays:
ƒ(x) = asin(bx - c) + d -2π 2π
-π π
Which is equivalent to:
ƒ(x) = AsinB(x - c/b) + D
The quot;starting point.quot;
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.

9. 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 =

10. 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
ƒ(x) = AsinB(x - C) + D

11. 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
ƒ(x) = AsinB(x - C) + D

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

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

14. Enter the values into your calculator, and use a sinusoidal regression to
determine the equation. Round the values of the parameters to one decimal
place. HOMEWORK
x -1 -0.5 0 0.5 1 1.5 2 2.5
y 1 -2.6 -5.6 -5.4 -2 1.4 1.6 -1.4