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

¼ Per 0 π π 3π 2π

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

-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

-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

-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

-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

+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

b

8 EX: • Graph y = sin 2x

Ref.no

Amp.1

Per. 2π/2 = π

¼ Per. π/4

St. Pt. 0

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. 0

Vert. Sh.none

0 1 0 -1 0

1 0 -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

0

none

19 y = tan (2x + π/2) y = tan 2 (x + /4)

Ref.Amp.Per.

¼ Per.

St. Pt.

Vert. Sh.

No128- 4none

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

22

-22↑

21 y = cot x

Ref.

Amp.

Per.

¼ Per.

St. Pt.

Vert. Sh.

No

1

4

0

none

22 y = 2 + cot x

Ref.

Amp.

Per.

¼ Per.

St. Pt.

Vert. Sh.

No

1

4

0

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