Introduction to transformations of the sine function: Amplitude and vertical shift.

1. The quot;periodicquot;
moments of our lives ...
or
Transformations of
the sine function
Sunshine Coast Panoramic by flickr user El Fotopakismo

2. The sine curve (graph) ... HOMEWORK

3. HOMEWORK

4. HOMEWORK

5. HOMEWORK

6. HOMEWORK

7. http://www.poodwaddle.com/worldclock.swf

8. 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 ND
Mean -17 -14 -6 4 12 17 20 18 12 6 -4 -14
Source: Winnipeg weather statistics

9. 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

11. Properties and Transformations of the sine function ...
Let's look at some graphs ...
http://fooplot.com

12. 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.

13. 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.

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

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

17. 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.

18. 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 =

19. 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