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Unit 6 – Introduction to Trigonometry
Graphing Sine and Cosine Functions
(Unit 6.4)
William (Bill) Finch
Mathematics DepartmentDenton High School
Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Lesson Goals
When you have completed this lesson you will:
I Graph the parent sine and cosine functions.
I Graph transformations of sine and cosine functions.
W. Finch DHS Math Dept
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Mapping the Unit Circle to the Sine Function
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Mapping the Unit Circle to the Cosine Function
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Periodic Function
A periodic function repeats its output values at regularintervals. These intervals are called the period of the function.
y = sin x Period = 2π
sin(x + n · 2π) = sin x
y = cos x Period = 2π
cos(x + n · 2π) = cos x
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
The Sine Function
Domain: (−∞, ∞)
Range: [−1, 1]
y-intercept: 0
Max: 1 at x = π2
+ n · 2πMin: −1 at x = 3π
2+ n · 2π
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
The Sine Function
Continuous: (−∞, ∞)
Symmetry: origin (oddfunction)
End Behavior: limx→−∞ sin xand limx→∞ sin x do notexist
Oscillation: −1 to 1
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
The Cosine Function
Domain: (−∞, ∞)
Range: [−1, 1]
y-intercept: 1
Max: 1 at x = n · 2πMin: −1 at x = π + n · 2π
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
The Cosine Function
Continuous: (−∞, ∞)
Symmetry: y -axis (evenfunction)
End Behavior: limx→−∞ sin xand limx→∞ sin x do notexist
Oscillation: −1 to 1
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
General Sinusoidal Model
y = a sin(bx + c) + d y = a cos(bx + c) + d
where a, b, c , and d are constants (a 6= 0, and b 6= 0)
Do you recall the roles of constants in the transformation offunctions?
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
“Key Points” for y = sin x
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
“Key Points” for y = cos x
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Vertical Dilations
Recall the constant a in y = af (x) produces a
vertical stretch when |a| > 1
vertical shrink when |a| < 1
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Amplitude of Sine and Cosine Functions
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 1
On the same axes, sketch two full periods of the graphs off (x) = sin x and g(x) = 3 sin x . Then identify the amplitudeof g .
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 2
On the same axes, sketch two full periods of the graphs of
f (x) = cos x and g(x) =1
4cos x . Then identify the amplitude
of g .
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Reflections wrt x-axis
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 3
On the same axes, sketch two full periods of f (x) = cos x andg(x) = −2.5 cos x . Then identify the amplitude of g .
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Horizontal Dilations
Recall the constant b in y = f (bx) produces a
horizontal shrink when |b| > 1
horizontal stretch when |b| < 1
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Period of the Sine and Cosine Functions
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 4
Sketch at least one full period of f (x) = sin x and
g(x) = sinπ
3x . Identify the period of g .
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 5
Sketch at least one full period of f (x) = cos x andg(x) = cos 2x . Identify the period of g .
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Frequency
Horizontal dilations affect the frequency of sinusoidalfunctions.
Frequency is thenumber of cyclescompleted in onehorizontal unit and is thereciprocal of period.
freq =1
period=|b|2π
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 6
Musical notes are classified by frequency. Modern orchestrastune to the frequency A 440 hertz (cycles per second). Writean equation for a sine function that can be used to model theinitial behavior of the sound wave if the amplitude is 0.2.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Horizontal Translations
Recall the constant c in y = f (x + c) produces a
horizontal shift left |c | units when c > 0
horizontal shift right |c | units when c < 0
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Phase Shift of Sine and Cosine Functions
A phase shift of a sinusoidal function is a horizontaltranslation of a sine or cosine function.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 7
State the amplitude, period, frequency, and phase shift of
y = 2 cos(
5x +π
4
). Then graph two periods of the function.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Vertical Translations
Recall the constant c in y = f (x) + c produces a
vertical shift up |c | units when c > 0
vertical shift down |c | units when c < 0
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Midline
The midline of a sinusoidal function is the horizontal linearound which the graph oscillates and is given by y = d whered is the vertical translation.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 8
State the amplitude, period, frequency, phase shift, andvertical shift of y = sin(x + π)− 2. Then graph two periods ofthe function.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 9
Write a sinusoidal function.
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
Example 10
Use the data below to create a sinusoidal model for theaverage monthly temperature (◦F ) at DFW.
Month (x) Temp (y)
Jan 1 57
Mar 3 69
May 5 84
Jul 7 96
Sep 9 89
Nov 11 67
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Introduction Periodic Parent Functions Amp Reflect Period Phase Shift Midline Applications Summary
What You Learned
You can now:
I Find values of trigonometric functions for any angle.
I Find the values of trigonometric functions using the unitcircle.
I Do problems Chap 4.4 #1-11 odd, 15-19 odd, 21, 30,31-34
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