1. Transformations of sine and cosine
Period, Amplitude, phase shift, and vertical shifts

2. Why?
Sine and cosine are periodic functions. Periodic functions are those whose outputs repeat regularly over time.
The outputs of sine and cosine oscilate symmetrically between some maximum and some minimum value.
In practical applications, the middle value is rarely zero. Often all outputs are positive. For example the output could be the height of a point on a wheel, or the water level at some point on a pier. The middle value would be the hub of the wheel, or the water level at slack tide.

3. More why
In practical applications, we may choose, or be forced to, start recording time, or measuring values when the output is something other than a maximum or middle value.
Changing the middle value to something other than zero is a vertical shift.
Starting the cycle at some point other than the middle or maximum value is a horizontal shift, called phase shift in science.
This week and next we will focus on the math, after break we will focus on the applications
Except for this brief reminder.

4. frequency
Frequency is the reciprocal of “period”, so the longer the period, the smaller (lower) the frequency, and the shorter the period, the larger (higher) the frequency.
Frequency: how many complete cycles within a given unit of time: usually seconds- so frequency units are hertz
Wavelength is directly proportional to the period:
The longer the wavelength the longer the period, and the smaller(lower) the frequency.

9. Radio telescope

10. Radio telescope array

11. Trig Curves Transformed
Graph Sine, Cosine Functions

12. Copyright © 2009 Pearson Education, Inc.

13. Variations of the Basic Graphs
We are interested in the graphs of functions in the form
y = A sin B (x – h) + k
and
y = A cos B (x – h) + k
B careful!
where A, B, h, and k are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.
Copyright © 2009 Pearson Education, Inc.

14. Trig Curves Transformed
Graph Sine, Cosine, and Tangent Functions
Equation of a Sine Function
Amplitude
Period
Complete Cycle

15. Trig Curves Transformed
Graphing a Sine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

16. Trig Curves Transformed
Graphing a Sine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

17. Trig Curves Transformed
Graphing a Sine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

18. Trig Curves Transformed
Graph Sine, Cosine, and Tangent Functions
Equation of a Cosine Function
Amplitude
Period
Complete Cycle

19. Trig Curves Transformed
Graphing a Cosine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

20. Trig Curves Transformed
Graphing a Cosine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

21. Trig Curves Transformed
Graphing a Cosine Functions
Draw one cycle of the function’s graph.
Amplitude
Period

22. Trig Functions Translation
Translations of Trigonometric Graphs
Translation of a Sine Function
Amplitude
Period

23. Trig Functions Translation
Translations of Trigonometric Graphs
Translation of a Sine Function
Amplitude
Period

24. Trig Functions Translation
Graphing a Vertical Translation
Graph the function.
Amplitude
Translation
down 2
Period

25. Trig Functions Translation
Graphing a Vertical Translation
Graph the function.
Amplitude
Translation
up 3
Period

26. Trig Functions Translation
Graphing a Horizontal Translation
Graph the function.
Amplitude
Period
Translation
right p

27. Trig Functions Translation
Graphing a Horizontal Translation
Graph the function.
Amplitude
Period
Translation
left p

28. Trig Functions Translation
Graphing a Horizontal Translation
Graph the function.
Amplitude
Period
Translation
right p/2

29. Trig Functions Translation
Graphing a Reflection
Graph the function.
Amplitude
Period
Reflection over x-axis

30. Trig Functions Translation
Graphing a Reflection
Graph the function.
Amplitude
Period
Reflection over x-axis