4.5 Graphs of Sine and 4.5 Graphs of Sine and Cosine FunctionsCosine Functions

AmplitudeAmplitude

PeriodPeriod

TranslationsTranslations

The graph of f(x)=sin x

Domain all Real numbers

Range -1 to 1

1,2

1,

2

3

The graph of f(x)=sin x

Sin x is an odd function

1,2

1,2

1,

2

3

1,2

3

Graph of f(x) = -sin x

Graph of the f(x) = Cos x

Domain: All real numbers

Range: -1 to 1

Graph of the f(x) = Cos x

Cos x is an even function

1, 1,

1,2 1,2

The red graph is Sin and the blue is Cos

xxfxxf

2sin)()cos()(

Amplitude changes the Range

Amplitude changes the Range

Since Amplitude is a distance, it is always positive.

To find it: the absolute value of the Maximum minus the Minimum divide by two.

Amp is written before the function

y = a*sin x

Amplitude changes the Range

Max 7; Min - 1 Amp.

1,2

7,2

42

)1(7

Period (wave length)

The distance before the function repeats its value. y = sin bx; here b is 1.

Period (wave length)

y = sin 2x b = 2period

b

2

1,4

1,4

5

Translation to move the graph of the function y = sin (x + c):

moves Right or Left

Here is

2sin

xy

Translation to move the graph of the functiony = sin x + d

moves up or down

Here is 2sin xy

With all the translations

The sine function is

f(x) = d + a sin b(x +c)

The cosine function is

f(x) = d + a cos b(x +c)

Applet for the Sine function

• http://www.analyzemath.com/trigonometry/sine.htm

HomeworkHomework

Page 307 – 310Page 307 – 310

##3, 9, 14, 20, 3, 9, 14, 20,

26, 32, 41, 49, 26, 32, 41, 49,

71, 7871, 78

HomeworkHomework

Page 307 – 310Page 307 – 310

## 7, 12, 17, 23, 7, 12, 17, 23,

27, 36, 45, 60, 27, 36, 45, 60,

76, 8676, 86