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Section 4.1 Basic Graphs. periodic function a function that repeats its values in regular intervals.

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Presentation on theme: "Section 4.1 Basic Graphs. periodic function a function that repeats its values in regular intervals."— Presentation transcript:

1 Section 4.1 Basic Graphs

2 periodic function a function that repeats its values in regular intervals

3 Examine each of the following graphs. Does the graph represent a periodic function?

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13 amplitude magnitude of change in the oscillating variable If the greatest value of f(x) is M and the least value of f(x) is m, then the amplitude of the graph of f(x) is M m

14 M m For trig functions, amplitude is distance to max or min value from the midline.

15 What is the amplitude of the periodic function? 1.75

16 What is the amplitude of the periodic function? 2

17 1.5

18 What is the amplitude of the periodic function? undefinednone

19 period smallest positive distance at which a function repeats For any function f(x), the smallest positive number p for which is called the period of f(x).

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23 What is the period of the periodic function? 6.5

24 What is the period of the periodic function? 3 

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26 1 Look for repeating y -values to determine period…but the period is the length along the x -axis.

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28 Sine Graph: f(  )= sin   sine

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30 Sine Graph: f(  )= sin 

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41 Amplitude: Period: Sine Graph: y = sin  Domain: Range: y -int: Graph of one cycle

42 Cosine Graph: f(  )= cos   cosine

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44  Cosine Graph: f(  )= cos 

45 Graph of one cycle

46 Cosine Graph: y = cos  Domain: Range: y -int: Amplitude: Period: Graph of one cycle

47  tangent Tangent Graph: f(  )= tan 

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67 Graph of one cycle is usually to

68 Amplitude: Period: Asymptotes: Tangent Graph: y = tan  Domain: Range: y -int: Graph of one cycle

69 Cosecant Graph: y = csc  To graph the cosecant, start with its reciprocal: sine. y = sin x

70 Cosecant Graph: y = csc  Where sin Ѳ = 0, csc Ѳ is undefined. These are asymptotes.

71 Cosecant Graph: y = csc  Where sin Ѳ = 1, csc Ѳ = 1. Where sin Ѳ = -1, csc Ѳ = -1.

72 Cosecant Graph: y = csc  Where sin Ѳ = ½ (  /6, 5  /6), csc Ѳ = 2. Where sin Ѳ = -½ (7  /6, 11  /6), csc Ѳ = -2.

73 Cosecant Graph: y = csc  Repeat for reciprocals of other y-values.

74 Amplitude: Period: Asymptotes: Cosecant Graph: y = csc  Domain: Range: y -int: Graph of one cycle

75 Secant Graph: y = sec  To graph the secant, start with its reciprocal: cosine. y = cos x

76 Secant Graph: y = sec  Where cos Ѳ = 0, sec Ѳ is undefined. These are asymptotes.

77 Secant Graph: y = sec  Add points where y = 1 and y = -1. Sketch rest of graph using asymptotes.

78 Amplitude: Period: Asymptotes: Secant Graph: y = sec  Domain: Range: y -int: Graph of one cycle

79 Cotangent Graph: y = cot  To graph the cotangent, you could start with its reciprocal: tangent.

80 Cotangent Graph: y = cot  Where tan Ѳ = 0, cot Ѳ is undefined. These are asymptotes. Where tan Ѳ is undefined, cot Ѳ = 0.

81 Cotangent Graph: y = cot  Reciprocals points are graphed.

82 Cotangent Graph: y = cot  Graph of one cycle To graph the cotangent, it may be easier to remember the asymptotes and change to a downward slope.

83 Cotangent Graph: y = cot  Domain: Range: y -int: Amplitude: Period: Asymptotes: Graph of one cycle

84 Things to notice: The period for tan and cot is π. For all other trig functions, it is 2 π. If the graph goes up or down to infinity, the amplitude is “undefined” or none. Asymptotes, and domain restrictions, are the same for tan and sec (π /2 + π k ). Asymptotes, and domain restrictions, are the same for cot and csc (π k ). All trig asymptotes have + π k, even when the period is 2 π.

85 The End.


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