Graphs of Trigonometric
Functions
Amplitude and Period of Sine and
Cosine Functions
Review!
•
Amplitude
•The amplitude of the function
y = A sin θ
and y = A cosθ
is the absolute
value of A or
A
•For the parent functions y = sin x
and y= cos x, the amplitude is 1.
Period of Sine and Cosine Functions
•The period of the function
y = sin kθ
and y = cos kθ is
2π
, where k > 0
k
•The period of the parent functions y=sin x
•And y=cos x is 2
π
Example # 1
• State the amplitude for the function
y = -2 sin x.
• Solution: The amplitude is 2. Why is it not -2?
• Graph y = -2 sin x and y = sin x on the same set
of axes.
• Compare
the graphs?
Compare and Contrast
1. How does the behavior of f (x) = A sin x compare
with the parent graph f (x) = sin x? How does the
A effect the graph? Similarities/differences?
2. How does the behavior of f (x) = sin kx compare
with the parent graph f (x) = sin x. How does the
k effect the graph? Similarities/differences?
3. So how does the behavior of f (x) = A sin kx
compare with the parent graph?
Example # 2
θ
1. State the period for the function y = cos 4 .
The period or cycle is 2π
θ
Graph y = cos 4
1
4
= 8π
and the parent function y = cos x.
Example # 3
•State the amplitude and period for the function
y = 5 cos 2x. Then graph the function.
•Solution: The amplitude is 5 and the period is
2π
=π
2
Example # 4
•Write and equation of the sine function with amplitude 2 and
period π
2
.
• Since the amplitude is 2 we know that A = 2 or -2.
So we have y=±2sinx
π
Since the period is
2
, we know that
So the equation of this sine function is
Y=±2sin4x
2π
π
=
2
k
which means
k=4
HW # 46
• Section 6-4
• Pp.373-377
• #17-27 all, 31, 32,
37, 39, 43, 49, 58,
62, 64