4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine.

4.1 Graphs of Sine and Cosine

OBJ: Graph sine and cosine

1 DEF: Sine Graph

1

0 π π 3π 2π

-1 2 2

0 1 0 -1 0

y = d + a(trig b (x + c))

a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2π

b

c (starting point)

d (vertical shift)

y = sin x

Ref.no

Amp.1

Per. 2π

¼ Per. π/2

St. Pt. 0

Vert. Sh.none

0 1 0 1 0

1 0

-1 π/2 3π/2 4π/2

2 DEF: Cosine Graph

0 π π 3π 2π

2 2

1 0 -1 0 1

2 DEF: Cosine Graph

- π 0 π π 3π 2π

2 2 2

1 0 -1 0 1

DEF: Periodic function

A function f with the property f(x) = f(x+p) for every real number x in the domain of f and for some real positive number p. The smallest possible positive value of p is the period of the function f.

3 EX: Graph y = 2 sin x

0 π π 3π 2π

2 2

0 1 0 -1 0

2(0 1 0 -1 0)

0 2 0 -2 0


0 π π 3π 2π

2 2

0 2 0 -2 0

DEF: Amplitude of Sine and Cosine

The graph of y = a sin x or y = a cos x will have the same shape as y = sin x or y cos x, respectively, except with range - a y a . The number a is called the amplitude.



b

c (starting point)

d (vertical shift)

4 y = -2 cos x

1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2

2 1 0 -1 π/2 3π/2

4π/2 -2

4 y = -2 cos x

Ref.yes

Amp.- 2

Per. 2π

¼ Per. π/2

St. Pt. 0

Vert. Sh.none

1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2

2 1 0

-1 π/2 3π/2 4π/2 -2

4 y = -2 cos x

2 1 0 -1 π/2 3π/2

4π/2 -2

DEF: Vertical Translation

A function of the form y =d + a sin b x or of the form y = d + a cos b x is shifted vertically when compared with y = a sin b x or y =a cos b x.



b

c (starting point)

d (vertical shift)

5 EX: Graph y = – 3 + 2 sin x

0 π π 3π 2π

2 2

1 DEF: Sine Graph

1

0 π π 3π 2π

-1 2 2

0 1 0 -1 0


0 π π 3π 2π

2 2

2(0 1 0 -1 0)

0 2 0 -2 0


1

0 π π 3π 2π

-1 2 2

2(0 1 0 -1 0)

0 2 0 -2 0


1

0 π π 3π 2π

-1 2 2

2(0 1 0 -1 0)

0 2 0 -2 0

-3-3-3 -3 -3


1

0 π π 3π 2π

-1 2 2

-3 -1 -3 -5 -3

DEF: Phase Shift

The function y = sin (x + c) has the shape of

the basic sine graph y = sin x, but with a

translation c units: to the right if c < 0

and to the left if c > 0. The number c is

the phase shift of the graph. The cosine

graph has the same function traits.

y = d + a(trig b (x + c)


b

c (starting point)

d (vertical shift)

EX: Graph y = sin (x – π/3)6 EX: Graph y = 4 – sin (x – π/3)

2 5 8 11 14 -1 6 6 6 6 6

0 1 0 -1 0

6 EX: Graph y = 4 – sin (x – π/3)

2 5 8 11 14 -1 6 6 6 6 6

0 1 0 -1 0

-1(0 1 0 -1 0

0 -1 0 1 0

6 EX: Graph y = 4 – sin (x – π/3)

2 5 8 11 14 -1 6 6 6 6 6

0 -1 0 1 0

+4 +4 +4 +4 +4

4 3 4 5 4

6 EX: Graph y = 4 – sin (x – π/3)

2 5 8 11 14 -1 6 6 6 6 6

4 3 4 5 4

EX: Graph y = 3cos (x + π/4)7 EX: Graph y =-3 + 3cos(x+π/4)


- 3 5 7 4 4 4 4 4

1 0 -1 0 1


- 3 5 7 4 4 4 4 4

1 0 -1 0 13(1 0 -1 0 1)

3 0 -3 0 3

7 EX: Graph y =-3 + 3cos(x+π/4)

- 3 5 7 4 4 4 4 4

3 0 -3 0 3

-3 -3 -3 -3 -3

0 -3 -6 -3 0

__ __ __ __ __ __ __ __ __

7 EX: Graph y =-3 + 3cos(x+π/4)

- 3 5 7 4 4 4 4 4

__ __ __ __ __ __ __ __ __

1 EX: Graph y = -2 +sin x

Ref, Amp

No, 1

Per

2 π

¼ Per 0 π π 3π 2π

π/2 2 2

St.Pt. 0

Vert. Shift

2

1 EX: Graph y = -2 +sin x

0 π π 3π 2π

2 2

0 1 0 -1 0

-2 -2 -2 -2 -2

-2 -1 -2 -3 -2

2 EX: Graph y = 3 – 2 cos x

Ref, Amp

Yes, -2

Per

2 π

¼ Per 0 π π 3π 2π

π/2 2 2

St.Pt. 0

Vert. Shift

3


0 π π 3π 2π

2 2

1 0 -1 0 1

-2(1 0 -1 0 1)

-2 0 2 0 -2


0 π π 3π 2π

-2(1 0 -1 0 1) 2 2

-2 0 2 0 -2

+3 +3 +3 +3 +3

1 3 5 3 1

4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine.

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Transcript of 4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine.