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Section 4.1 Basic Graphs

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periodic function a function that repeats its values in regular intervals

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

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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

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

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What is the amplitude of the periodic function? 1.75

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What is the amplitude of the periodic function? 2

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1.5

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What is the amplitude of the periodic function? undefinednone

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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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What is the period of the periodic function? 6.5

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What is the period of the periodic function? 3

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2

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

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Sine Graph: f( )= sin sine

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Sine Graph: f( )= sin

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

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Cosine Graph: f( )= cos cosine

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Cosine Graph: f( )= cos

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Graph of one cycle

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

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tangent Tangent Graph: f( )= tan

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Graph of one cycle is usually to

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

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

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

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

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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.

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

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

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

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

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

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

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

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

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Cotangent Graph: y = cot Reciprocals points are graphed.

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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.

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

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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.

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The End.

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