4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the...

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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).

Transcript of 4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the...

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).

4.5 – Graphs of Sine and Cosine

x y

0 0

1

π 0

−1

2π 0

y = sin x

23π

−2π −π π 2π 3π 4π

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

4.5 – Graphs of Sine and Cosine

x y

0 1

0

π −1

0

2π 1

y = cos x

23π

−2π −π π 2π 3π 4π

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

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

b

c ,

b

c

b

b

c

4.5 – Graphs of Sine and Cosine

Graph the equation y = 3 sin(2x − π)

amplitude:

period:

phase shift:

interval:

3

= π2

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

4.5 – Graphs of Sine and Cosine

Graph the equation y = sin(2x − π) + 2

amplitude:

period:

phase shift:

interval:

vertical shift:

1

= π

up 2 units

2

2

π

2

3π ,

2

π

π 2π2π

23π