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Warm Up

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Chapter 4 Graphs of the Circular Functions Section 4.1 Graphs of the Sine and Cosine Functions Objective: SWBAT graph the sine and cosine functions with variations in amplitude and periods.

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Periodic Functions Many things in daily life repeat with a predictable pattern, such as weather, tides, and hours of daylight. This periodic graph represents a normal heartbeat. A function that repeats itself after a specific period of time is called a Periodic Function. Sine and Cosine functions are periodic functions.

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Sine and Cosine Functions We are going to deconstruct the Unit circle and graph the sine and cosine functions on graph… Remember: Cosine is in the “x” spot in an ordered pair and Sine is in the “y” spot.

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Vocabulary Sin wave - is a repetitive change or motion which, when plotted as a graph, has the same shape as the sine function.sine function Amplitude - is the maximum distance it ever reaches from zero. Period - is the time it takes to perform one complete cycle. (2 π for cosine and sin)

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Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. (5 total) y x sin x 0x 010 0

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Sine Function f ( x ) = sin x The graph is continuous over its entire domain, (– , ). Its x -intercepts are of the form n , where n is an integer. Its period is 2 . The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin(– x ) = –sin( x ).

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Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. Amplitude The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [ |a|, |a|]. The amplitude is |a|.

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Example: Graph y = 3 sin x compare to y = sin x. Make a table of values. 0 33 0303sin x 0 11 010sin x 3 /2 /2 0x Amplitude The range of y = 3sin x is [–3, 3].

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Amplitude y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = sin x y = 2 sin x Amplitude

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Period For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period Divide the interval into four equal parts to obtain the values for which sin bx equal –1, 0, or 1. (These values give the minimum points, x-intercepts, and maximum points on the graph)

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GRAPHING y = sin bx Graph y = sin 2 x and compare to the graph of y = sin x. The coefficient of x is 2, so b = 2, and the period is The endpoints are 0 and and the three points between the endpoints are The x -values are:

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GRAPHING y = sin bx Graph y = sin 2 x and compare to the graph of y = sin x. Y = sin2xXY(x, y) 0

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GRAPHING y = sin bx Graph y = sin 2 x and compare to the graph of y = sin x.

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Homework Page 141 # 1-7 (odds) #10

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