4.5 Notes Honors Precalculus
Date: __________________________________
I. Definitions Related to Sine and Cosine Curves
Sine Curve
Cosine Curve
Periodic Function
Cycle
Period
Amplitude
Key Points
II. Basic Graph of Sine Curve
x =θ
y = sin θ
−2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
y = sin θ
Domain:
Range:
Five key points:
Max:
Min:
Intercepts:
2π
III. Basic Graph of Cosine Curve
−2π
x =θ
y = cosθ
−
−π
3π
2
−
π
2
0
π
2
π
3π
2
2π
y = cosθ
Domain:
Range:
Five key points:
Max:
Min:
Intercepts:
IV. Translations of Sine and Cosine Curves
The graphs of y = a sin(bx) and y = a cos(bx) , have the following characteristics. (Assume
b > 0 .)
Amplitude & Period
of Sine & Cosine
* ___________ is the amplitude of the function
Functions
*____________ is the period of the function
Ex. 1
Graph y = 3sin x for two periods.
Ex. 2
Graph y = cos(2x) for one period.
Ex. 3
5
x
Graph y = − cos( ) for one period.
2
2
Ex. 4
Graph y =
1
π
sin( x) for one period.
2
3
The graphs of y = a sin(bx − c) + d and y = a cos(bx − c) + d , have the following characteristics.
(Assume b > 0 .)
Phase Shift &
Vertical Shift of
Sine & Cosine
Functions
* Phase Shift: The left and right endpoints of one-cycle interval can be determined by solving the
equations bx − c = __________ and bx − c = __________.
* Vertical Shift: ____________ determines the vertical translation
Ex. 5
Graph y = sin(2x − π ) for one period.
Ex. 6
Graph y = −2cos(2π x + π ) + 3 for one period.