14.1 Graphing Sine, Cosine and Tangent Functions

14.1 Graphing Sine, Cosine and Tangent Functions

14.1 Graphing Sine, Cosine and Tangent Functions Algebra 2 The graph of y=sin x The graph of y=cos x Characteristics of y=sinx and y=cosx The

domain is all real numbers The range is -1 y 1 The function is periodic-the graph has a repeating pattern. (Shortest repeating pattern called a cycle and the horizontal length is called the period) Both have a period of 2. Characteristics of y=sinx and y=cosx The

maximum value of y=sinx is x 2 n M=1 and occurs when 2 The maximum value of y=cosx is M=1 and occurs when x=2n. The minimum value of y=sinx is 3 x

at 2 n m=-1 and occurs 2 The minimum value of y=cosx is m=-1 and occurs when x=(2n+1) Characteristics of y=sinx and y=cosx The is

amplitude of both functions 1 M m 1 2 Amplitude the graph. is half the height of Characteristics of y=a sin bx

and y=a cos bx The amplitude and period of the graphs of y = a sin bx and y = a cos bx where a and b are nonzero numbers are as follows. a amplitude= 2 period=b Examples:

Graph the functions 1 y cos x 2 1 y sin x 2 y 2 sin x

4 Examples: Give the amplitude, period. And five key points of the graph of each function. y sin x y 3 cos x

y 2 sin x 4 Definition Frequency- the number of cycles per unit of time (frequency is the reciprocal of the period) Examples:

A tuning fork vibrates with frequency f=880 hertz (cycles per second.) You strike the tuning fork with a force that produces a maximum pressure of 4 pascals. Write a sine model that gives the pressure P as a function of t (in seconds). Graph the model. Examples:

You pluck the string of a violin so that it vibrates with frequency f = 660 hertz (cycles per second.) The force of the pluck produces a maximum pressure of 2 pascals. Write a sine model that gives the pressure P as a functions of time t (in seconds). Then give the amplitude and period of the function's graph.

Tangent Functions The graph of y=tanx has the following characteristics. The domain is all real numbers except odd multiples of . At , the 2 2 graph has vertical asymptotes

The range is all real numbers. The graph has a period of Characteristics of y=a tan bx If a and b are nonzero real numbers, the graph of y= a tan bx has these characteristics. The period is b

There are vertical asymptotes at odd multiples of 2b Examples: Graph the functions. y 3 tan x 2

y 1 tan 1 x 2 3 Question How do you find the amplitude, period and vertical asymptotes of a sine, cosine, or tangent function from its equation?

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