November 06, 2014
Section 4.5 Graphs of Sine and Cosine Functions
Objective: Sketch graphs of sine and cosine functions and graphs of translations
of these functions.
y = sin x
Key Points
One Period or Cycle (2π)
Note: Be sure calculator is in radian mode when graphing.
y = cos x
Key Points
One Period or Cycle (2π)
5 Key Points
y = sinx
y = cosx
November 06, 2014
Ex: Sketch the following...
a) y = 2 sin x
2π
3π
2
π
π
2
π
2
π
3π
2
b) y = 0.4 cos x
2π
2π
3π
2
π
2
π
π
2
π
3π
2
2π
*In the equation y = a sin x and y = a cos x, a is the amplitude
(maximum displacement from equilibrium) ...works as a scaling
factor.
a > 1 --> vertical stretch
a < 1 --> vertical shrink
Ex: Sketch the graph of y = cos 2x
y = cos x
y = cos 2x
2π
3π
2
π
π
2
π
2
π
3π
2
2π
(0, 1)
π
( 2 , 0)
(π, -1)
π
π
( 3π
, 0) (2π, 1)
2
3π
(0, 1) ( 4 ,0) ( 2 , -1) ( 4 , 0) (π, 1)
*Tip: Add "period" from left
endpt. to move right.
4
period of
find the
e
w
o
d
ad on...
How
x? Re
2
s
o
c
=
y
Check this out!
*Since y = a cos x completes a period from x=0 to x=2π, it
follows that y = a cos bx completes one period from x=0/b=0
to x = 2π/b.
The period of y = a sin bx and y = a cos bx is 2π/b.
0 < b < 1 --> Horizontal stretch
b > 1 --> Horizontal shrink
November 06, 2014
General equations:
y = d + a sin (bx-c)
y = d + a cos (bx-c)
Amplitude ( a ) - maximum displacement from equilibrium
Vertical shift (d)
Phase shift (c/b) - horizontal shift
Period: 2π/b
Curve oscillates about horizontal line y = d
Left and right endpoints of a one-cycle interval can be
determined by solving the equations bx-c = 0 and bx-c = 2π.
If a < 0, then the graph is a reflection in the x-axis.
Ex: Find the amplitude, period, and phase shift of y = 2sin(x - π/4).
Amplitude = 2; Perio
Ex: Find the five key points.
(π/4, 0), (3π/4, 2),
Ex: Find a trig. function to model the data in the following table.
2π
x
0
π/2
π
3π/2
2π
y
2
4
2
0
2
3π
2
π
π
2
π
2
π
3π
2
2π
y = 2sin x + 2 or
y = 2 + 2sinx