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Unit 4 Section 4

# Unit 4 Section 4 - 4.4 GRAPHS OF CIRCULAR FUNCTIONS Graphs...

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4.4 GRAPHS OF CIRCULAR FUNCTIONS Graphs of circular functions are easier to sketch compared to the graphs of algebraic functions since they have properties that make their graphs more “predictable”. Some of these properties include period, amplitude, and phase shifts. Properties of the Graph of Circular Functions 1. Circular functions are periodic functions. This means that their function values repeat after a fixed interval. The fixed interval p is called the period (or wavelength ) . This property can be observed in their graphs. A cycle is simply the portion of a graph of a function f over an interval of length equal to the period of f. 2. The amplitude a (or maximum function value) of a graph of a periodic function refers to one- half the absolute difference between the highest and lowest function values. 3. Phase shift is a horizontal (or vertical) shift from the standard graph. Graph of the Sine and Cosine Functions The sine function defined by has the following properties: The domain is . The range is . The amplitude is equal to 1. There is no phase shift. Figure 1. The Sine Curve

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The cosine function defined by has the following properties: The domain is . The range is . The amplitude is equal to 1. There is no phase shift. Figure 2. The Cosine Curve General Form of the Sine and Cosine Functions The general forms of the sine and cosine functions are as follows: where a, b, h and k are constants and a, b are nonzero. PROPERTIES: 1. The amplitude of the graphs of the sine and cosine functions are given by . 2. The range is also affected by the value of , that is, . 3. The period of the graph is given by the formula . 4. The horizontal phase shift is determined by the constant . If , then the graph shifts units to the right. If , then the graph shifts units to the left. 5. The vertical phase shift is determined by the constant . If , then the graph shifts units upward. If , then the graph shifts units downward.
Sketching One Cycle of the Graph To sketch a cycle of the graph of the sine and cosine functions, do the following: 1. Identify values for a, b, k , and h. 2. Determine the amplitude, range, period and phase shift of the function.

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