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The concept of periodic functions, focusing on the period, amplitude, and graphical representation of sine and cosine functions. It provides examples of graphing these functions and translating them using phase shift. Students will learn how to identify key points on the graph and understand the role of amplitude, period, phase shift, and vertical shift in shaping the graph.
Typology: Lecture notes
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Period Functions The graph of a periodic function shows a repeating pattern. The distance from 1 point on the graph to the point where the pattern begins repeating is called the period.
Periodic Function : repeats a pattern of y-values at regular intervals Cycle : one complete pattern, a cycle may begin at any point on the graph Period : the horizontal (x) length of one cycle Amplitude : half the distance between the maximum and minimum y- values of a function
Examples:
Graphing Sine Sine Function: y = sinx (amplitude =1, period = 2π)
We will graph the angle measure (the x value) in radians. To graph by hand we will find 5 key points. These points are the maximum, the minimum, and the x-intercepts. We will usually graph only 1 cycle.
The graph of a sine function is called a sine curve.
For y = a sin bx with a≠ 0, b >0 and x in radians :
2 (^) b is the period of the function
| a | = ½, so the amplitude is ½ b = 2 so there are 2 cycles from 0 to 2π
Divide the period into fourths. Using the values of the amplitude and period plot the pattern zero-max-zero-min-zero.
Translating Sine and Cosine Functions
Phase Shift: a horizontal translation of a periodic function.
For y = a sin b(x-h) + k or y = a cos b(x-h) + k
2 (^) b is the period of the function
| a | =1, so the amplitude is 1 b = 2 so there are 2 cycles from 0 to 2π
Use the 5 key points.
Since h = 3 and k =
3 2 translate
the graph 3 units to the right and
3 2 units down. Sketch the graph.
