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:

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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 )

Example 4

A bass tuba can hit a note with a frequency

of 50 cycles per second (50 hertz) and

an amplitude of 0.75.

Write an equation for a

cosine function that

can be used to

model the initial

behavior of

the sound

wave associated

with the note.

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) + 1. Then graph two periods of the function.
Assignment

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