November 29, 2012
Period and Amplitude of the Sine and
Cosine Functions
Warm-up:
Finish Sine and Cosine Function Activity (15 minutes)
HW 4.5: Pg. 328-329
#1-6, 35-43
Lesson 4.5 The Sine Function
ϴ sinϴ
0
π/2
Π
3π/2
2π
1
π/2
-1
π
ϴ
3π/2
2π
The Cosine Function
ϴ
cosϴ
0
π/2
Π
3π/2
2π
1
π/2
-1
π
ϴ
3π/2
2π
Sketch the sine and cosine curve from -2π to 2π
Always use the intercepts, maximum, and minimum
points to sketch sine or cosine curve.
y = sin x
θ
y = cos x
θ
Finding Amplitude of Sine and Cosine Graphs
The amplitude of the graphs is how high and low
your graph goes from the x-axis. *The range of
the function.
What is the amplitude for y = sinx and y = cosx?
|a| is the magnitude or the amplitude of the graph.
y = a sinθ or
y = a cosθ
Example: What is the amplitude of y = 3 cosx ?
Graphing with different amplitudes
Graph y = 2 sinx
1st: graph the zeroes
2nd: graph max and min
2
1
ϴ
-1
-2
π/2
π
3π/2
2π
Sketch a graph of each, then identify the
intercepts, max, and min.
1
y = sin x
2
Vertical Shrink
π/2
π
3π/2
ϴ
2π
y = -2cos x
π/2
π
3π/2
2π
ϴ
Vertical Stretch
Finding the Period of Sine and Cosine Graphs
The period of the sine and cosine curves is the
angle at which the curve repeats its pattern.
The period changes if you multiply the angle by
some factor, b:
y = sin(bϴ)
y = cos(bϴ)
2p
Period of sine or cosine =
b
What is the period of y = cos(2x)?
Graph y = cos(2x), include 2 full periods
1) Identify the period and amplitude.
2) Divide the period into four parts (these will be our
zeros, max and mins since sine and cosine always follow
the same pattern!)
3) Use your answer from 2 to scale your x-axis
Horizontal Shrink
x
Graph y = 3sin , include 2 full periods
2
1) Identify the period and amplitude.
2) Divide the period into four parts (these will
be our zeros, max and mins since sine and
cosine always follow the same pattern!)
3) Use your answer from 2 to scale your x-axis
Horizontal Stretch
Start HW 4.5 #1 and #37, 39
1. Find the period and amplitude:
y = 3 sin 2x, then sketch
37.Sketch the graph y = 1 cos x
3
39. Sketch the graph
x
y = cos
2