# 4-4 Graphing Sine and Cosine

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4-4 Graphing Sine and Cosine
Chapter 4 Graphs of Trigonometric Functions

Warm-up Find the exact value of each expression. sin 315° cot 510°

6-3 Objective: Use the graphs of sine and cosine (sinusoidal) functions
6-4 Objectives: Find amplitude and period for sine and cosine functions, and Write equations of sine and cosine functions given the amplitude and period. Graph transformations of the sine and cosine functions

Recreate the sine graph.
Domain and Range x- and y-intercepts symmetry

Recreate the cosine graph.
Domain and Range x- and y-intercepts symmetry

KeyConcepts: Transformations of Sine and Cosine Functions
For y = a sin (bx + c) + d and y = a cos (bx + c) + d, Amplitude (half the distance between the maximum and the minimum values of the function or half the height of the wave) = |a|

Example 1 Describe how the graphs of f(x) = sin x and g(x) = 2.5 sin x
are related. Then find the amplitude of g(x). Sketch two periods of both functions.

Example 2 Reflections Describe how f(x) = cos x and g(x) = -2cos x are related. Then find the amplitude of g(x). Sketch two periods of both functions.

KeyConcepts: Transformations of Sine and Cosine Functions
For y = a sin (bx + c) + d and y = a cos (bx + c) + d, Period (distance between any two sets of repeating points on the graph) =

Example 3 Describe how the graphs of f(x) = cos x and g(x) = cos are related. Then find the period of g(x). Sketch at least one period of both functions.

KeyConcepts: Transformations of Sine and Cosine Functions
For y = a sin (bx + c) + d and y = a cos (bx + c) + d, Frequency (the number of cycles the function completes in a one unit interval) = (note that it is the reciprocal of the period or )

A bass tuba can hit a note with a frequency of 50 cycles per second (50 hertz) and an amplitude of Write an equation for a cosine function that can be used to model the initial behavior of the sound wave associated with the note. Example 4

KeyConcepts: Transformations of Sine and Cosine Functions
For y = a sin (bx + c) + d and y = a cos (bx + c) + d, Phase shift (the difference between the horizontal position of the function and that of an otherwise similar function) =

Example 5 State the amplitude, period, frequency, and phase shift of Then graph two periods of the function.

KeyConcepts: Transformations of Sine and Cosine Functions
For y = a sin (bx + c) + d and y = a cos (bx + c) + d, Vertical shift (the average of the maximum and minimum of the function) = d (Note the horizontal axis—the midline–is y = d)

Example 6 State the amplitude, period, frequency, phase shift, and vertical shift of y = sin (x + π) Then graph two periods of the function.

Assignment P. 264, 1, 3, 9, 15, 17, 19.