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 π.

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

Characteristics of y=sinx and y=cosx The amplitude of both functions is Amplitude is half the height of the graph.

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. amplitude= period=

Examples: Graph the functions ◦

Examples: Give the amplitude, period. And five key points of the graph of each function. ◦

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 graph has vertical asymptotes ◦ The range is all real numbers. ◦ The graph has a period of π

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 ◦ There are vertical asymptotes at odd multiples of

Examples: Graph the functions. ◦

Question How do you find the amplitude, period and vertical asymptotes of a sine, cosine, or tangent function from its equation?