Transcript 4.5 Graphs of Sine and Cosine Functions

4.5 Graphs of Sine and
Cosine Functions
Amplitude
Period
Translations
The graph of f(x)=sin x
Domain all Real numbers
Range -1 to 1
 
 ,1
2 
 3

 ,1
 2

The graph of f(x)=sin x
Sin x is an odd function
 
 ,1
2 
  3 
,1

 2

 

,1

 2

 3

 ,1
 2

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
 2 ,1
2 ,1
  ,1
 ,1


f ( x)  cos( x)  f ( x)  sin   x 
2

The red graph is Sin and the blue is 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.
 
 ,7 
2 
 

,1

 2

7  (1)
4
2
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 = 2
 
 ,1
4 
 5 
,1

 4 
2
 period
b
Translation to move the graph of the
function
y = sin (x + c):
moves Right or Left
Here is


y  sin  x  
2

Translation to move the graph of the
function y = sin x + d
moves up or down
Here is
y  sin x  2
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
Homework
Page 307 – 310
#3, 9, 14, 20,
26, 32, 41, 49,
71, 78
Homework
Page 307 – 310
# 7, 12, 17, 23,
27, 36, 45, 60,
76, 86