4.3 Period Changes and Graphs other Trig Functions Obj: Graph sine and cosine with period changes Obj: Graph other Trig Functions

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

2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π 2 2 1 0 -1 0 1 0 π π 3π 2π 2 2 1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2

2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π -2(1 0 -1 0 1) 2 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

2 EX:  Graph y = 3 – 2 cos x 0 π π 3π 2π -2(1 0 -1 0 1) 2 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

3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 1 0 -1 0

3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 1 0 -1 0

3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 1 0 -1 0 0 π π 3π 2π -1 2 2 0 1 0 -1 0 -2/3(0 1 0 -1 0) 0 -2/3 0 2/3 0

3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2 0 -2/3 0 2/3 0 0 π π 3π 2π -1 2 2 0 -2/3 0 2/3 0 +½ +½ +½ +½ +½ ½ -1/6 ½ 5/6 ½

3 EX:  Graph y =  –  sinx 1 0 π π 3π 2π -1 2 2

4 EX:  Graph y = 4 cos (x – π) 1 07π 10π 13π 16π 19 -1 6 6 6 6 6 1 0 -1 0 1

4 EX:  Graph y = 4 cos (x – π) 1 0 7π 10π 13π 16π 19 -1 6 6 6 6 6 1 0 -1 0 1

4 EX:  Graph y = 4 cos (x – π) 1 0 7π 10π 13π 16π 19 -1 6 6 6 6 6 1 0 -1 0 1 4(1 0 -1 0 1) 4 0 -4 0 4

4 EX:  Graph y = 4 cos (x – π) 1 0 7π 10π 13π 16π 19 -1 6 6 6 6 6

5 EX:  Graph y = 1 +sin(x + π/6) - 2π 5π 8π 11π 6 6 6 6 6 0 1 0 -1 0 ½(0 1 0 -1 0) 0 ½ 0 -½ 0

5 EX:  Graph y = 1 +sin(x + π/6) - 2π 5π 8π 11π 6 6 6 6 6 0 ½ 0 -½ 0 +1 +1 +1 +1 +1 1 1½ 1 ½ 1

5 EX:  Graph y = 1 +sin(x + π/6) - 2π 5π 8π 11π 6 6 6 6 6 0 ½ 0 -½ 0 +1 +1 +1 +1 +1 1 1½ 1 ½ 1

4.1 Period changes in graphs of Sine and Cosine OBJ: Find the period for a sine and cosine graph

y = d + a(trig b (x + c) a (amplitude) multiply a times (0 |1 0 -1 0 1) Sin|Cos -a Reflection b (period) 2π b can be factored out OR c (starting point) Set (bx + __) = 0 instead of completing factoring with b d (vertical shift)

DEF:  Period of Sine and Cosine The graph of y = sin b x will look like that of sin x, but with a period of  2  .  b  Also the graph of y = cos b x looks like that of y = cos x, but with a period of  2 

8 EX: • Graph y = sin 2x Ref. no Amp. 1 Per. 2π/2 = π ¼ Per. π/4 St. Pt. Vert. Sh. none

8 EX: • Graph y = sin 2x  2 3 4 -1 4 4 4 4 0 1 0 -1 0

8 EX: • Graph y = sin 2x  2 3 4 -1 4 4 4 4

8 EX: • Graph y = sin 2x Ref. no Amp. 1 Per. 2π/2 = π ¼ Per. π/4 St. Pt. Vert. Sh. none 0 1 0 -1 0 1 -1 π/4 3π/4 4π/4

EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x  2 3 4 -1 6 6 6 6

EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x  2 3 4 -1 6 6 6 6 1 0 -1 0 1

EX: • Graph y = -2cos 3x EX 9 • Graph y = 3 – 2cos 3x  2 3 4 -1 6 6 6 6 -2(1 0 -1 0 1) -2 0 2 0 -2

 2 3 4 -1 6 6 6 6 -2 0 2 0 -2 +3 +3 +3 +3 +3 1 3 5 3 1

10 EX: Graph y = –2cos3(x+π) 3

10 EX: Graph y = –2cos3(x+π) 3 -2 -  2 6 6 6 6

10 EX: Graph y = –2cos3(x+π) 3 -2 -  2 6 6 6 6

10 EX: Graph y = –2cos3(x+π) 3 -2 -  2 6 6 6 6

11 EX: • Graph y = cos(2x/3)

11 EX: • Graph y = cos(2x/3) 1 0 3π 6π 9π 12π -1 4 4 4 4

11 EX: • Graph y = cos(2x/3) 1 0 3π 6π 9π 12π -1 4 4 4 4

11 EX: • Graph y = cos(2x/3) 1 0 3π 6π 9π 12π -1 4 4 4 4

12 EX: Graph y = –2 sin 3x

12 EX: Graph y = –2 sin 3x 1 0 π 2π 3π 4π -1 6 6 6 6

12 EX: Graph y = –2 sin 3x 1 0 π 2π 3π 4π -1 6 6 6 6

12 EX: Graph y = –2 sin 3x 1 0 π 2π 3π 4π -1 6 6 6 6

13 EX: Graph y = 3 cos ½ x

13 EX: Graph y = 3 cos ½ x 1 0 π 2π 3π 4π -1

13 EX: Graph y = 3 cos ½ x 1 0 π 2π 3π 4π -1

13 EX: Graph y = 3 cos ½ x 1 0 π 2π 3π 4π -1

4.2 Graphs of the Other Trigonometric Functions OBJ: Graph Other Trigonometric Functions

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)

Graph y = cos x 0 π π 3π 2π 2 2

14 EX: Graph y = sec x 0 π π 3π 2π 2 2

14 EX: Graph y = sec x 0 π π 3π 2π 2 2

15 EX: Graph y = 2 + sec(2x–π)

15 EX: Graph y = 2 + sec2(x–π) 2 0 2π 3π 4π 5π 6 4 4 4 4 4

15 EX: Graph y = 2 + sec2(x–π) 2 0 2π 3π 4π 5π 6 4 4 4 4 4

Graph y = sin x 1 0 π π 3π 2π -1 2 2

16 EX: Graph y = csc x 1 0 π π 3π 2π -1 2 2

17 EX: Graph y = csc (x + π/3)

17 EX: Graph y = csc (x + π/3) -2π -π 0 π 2 6 6 6 6

17 EX: Graph y = csc (x + π/3) -2π -π 0 π 2 6 6 6 6

17 EX: Graph y = csc (x + π/3) -2π -π 0 π 2 6 6 6 6

18 y = tan x Ref. Amp. Per. ¼ Per. St. Pt. Vert. Sh. No 1  4 none

19 y = tan (2x + π/2) y = tan 2 (x + /4) Ref. Amp. Per. ¼ Per. St. Pt. Vert. Sh. No 1  2 8 - 4 none

20 y = 2 + ¼ tan (½x + π) y=2+¼ tan½(x + 2 π) Ref. Amp. Per. ¼ Per. St. Pt. Vert. Sh. No ¼ 2  2 -2 2↑

21 y = cot x Ref. Amp. Per. ¼ Per. St. Pt. Vert. Sh. No 1  4 none

22 y = 2 + cot x Ref. Amp. Per. ¼ Per. St. Pt. Vert. Sh. No 1  4 2↑

6 EX: Graph y =-3 – 2cos(x+5π/6)

6 EX: Graph y =-3 – 2cos(x+5π/6) -5 -2π π 4π 7π 6 6 6 6 6

6 EX: Graph y =-3 – 2cos(x+5π/6) -5 -2π π 4π 7π 6 6 6 6 6

6 EX: Graph y =-3 – 2cos(x+5π/6) -5 -2π π 4π 7π 6 6 6 6 6