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

1 DEF:  Sine Graph 1 0ππ3π 2π

1 DEF:  Sine Graph 1 0ππ3π 2π

y = d + a (trig b ( x + c) ) a (amplitude) multiply a times (0 | ) 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 π/2 3π/2 4π/2

2 DEF:  Cosine Graph 0ππ3π 2π

2 DEF:  Cosine Graph - π 0ππ3π 2π

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 0ππ3π 2π ( )

3 EX:  Graph y = 2 sin x 0ππ3π 2π

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.

y = d + a(trig b ( x + c) ) a (amplitude) multiply a times (0 | ) b (period) 2π b c (starting point) d (vertical shift)

4 y = -2 cos x ( ) 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 ( ) π/2 3π/2 4π/2 -2

4 y = -2 cos x 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.

y = d + a(trig b ( x + c) ) a (amplitude) multiply a times (0 | ) b (period) 2π b c (starting point) d (vertical shift)

5 EX:  Graph y = – sin x 0ππ3π 2π 2

1 DEF:  Sine Graph 1 0ππ3π 2π

3 EX:  Graph y = 2 sin x 0ππ3π 2π 2 2( )

5 EX:  Graph y = – sin x 1 0ππ3π 2π ( )

5 EX:  Graph y = – sin x 1 0ππ3π 2π ( )

5 EX:  Graph y = – sin x 1 0ππ3π 2π

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

6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14  (

6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14 

6 EX:  Graph y = 4 – sin (x – π/3) 2  5  8  11  14 

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

-   3  5  7 

EX:  Graph y = 3cos (x + π/4) 7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  ( )

7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  __ __ __ __ __ __ __ __ __

7 EX: Graph y =-3 + 3cos(x+π/4) -   3  5  7  __ __ __ __ __ __ __ __ __

1 EX:  Graph y = -2 +sin x Ref, Amp No, 1 Per 2 π ¼ Per 0ππ3π 2π π/2 2 2 St.Pt. 0 Vert. Shift 2

1 EX:  Graph y = -2 +sin x 0ππ3π 2π

1 EX:  Graph y = -2 +sin x 0ππ3π 2π

2 EX:  Graph y = 3 – 2 cos x Ref, Amp Yes, -2 Per 2 π ¼ Per 0ππ3π 2π π/2 2 2 St.Pt. 0 Vert. Shift 3

2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π ( )

2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π -2( )

2 EX:  Graph y = 3 – 2 cos x 0ππ3π 2π -2( )