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4.5 – Graphs of Sine and Cosine
A function is periodic if
f(x + np) = f(x)
for every x in the domain of f,every integer n,
and some positive number p (called the period).
Characteristics of the Sine Function1) The domain is .
2) The range is [-1, 1].
3) The period is .
4) The sine function is an odd function. It is symmetric with respect to the origin.
sin (x) = -sin (x)
,
2
Characteristics of the Cosine Function1) The domain is .
2) The range is [-1, 1].
3) The period is .
4) The sine function is an even function. It is symmetric with respect to the y-axis.
cos (x) = cos (-x)
,
2
4.5 – Graphs of Sine and Cosine
−2π −π π 2π 3π 4π
Graphing y = a sin x
y = 2 sin x
y = sin x
y = sin x½
4.5 – Graphs of Sine and Cosine
−2π −π π 2π 3π 4π
Graphing y = sin bx
y = sin 2x
y = sin x
y = sin x½
4.5 – Graphs of Sine and Cosine
−2π −π π 2π 3π 4π
Graphing y = sin(x − c)
y = sin(x + 2)y = sin x
y = sin(x − π)
π
4.5 – Graphs of Sine and Cosine
Graphing y = sin(bx − c)
0 bx − c 2π
c bx c + 2π
starting point ending point
b
2π
b
cx
b
c
4.5 – Graphs of Sine and Cosine
Graphing y = a sin(bx − c)
1. Find amplitude = | a |
2. Find period =
3. Find phase shift =
4. Find the interval on the x-axis.
5. Divide the interval into fourths to plot “key points”.
6. Graph one period. Extend if necessary.
b
2π
b
c ,
b
c
b
2π
b
c
4.5 – Graphs of Sine and Cosine
Graph the equation y = 3 sin(2x − π)
amplitude:
period:
phase shift:
interval:
3
= π2
2π
2
π
2
3π ,
2
π
π 2π2π
23π
4.5 – Graphs of Sine and Cosine
Graph the equation y =
amplitude:
period:
p. s.:
interval:
π
2
xcos
2
1
4π2π
212π
2
1
= 2π(2) = 4π
21π
= −π(2) = −2π
[−2π , 2π]
−2π
1
−1
4.5 – Graphs of Sine and Cosine
−2π −π π 2π 3π 4π
Graphing y = sin(x) + d
y = sin(x) + 2 y = sin x
y = sin(x) − 1