Chapter 14 Trigonometric Graphs and Identities14.1 Graphing Trig Functions

Graphs of Sine and Cosine FunctionsGraphs of Sine and Cosine Functions

Objectives:Objectives:

• to sketch the graphs of basic sine and cosine functions;to sketch the graphs of basic sine and cosine functions;• to use amplitude and period to help sketch the graphs of sine to use amplitude and period to help sketch the graphs of sine

and cosine functions;and cosine functions;• to sketch translations of the graphs of sine and cosine to sketch translations of the graphs of sine and cosine

functions;functions;• to use sine and cosine functions to model real-life data.to use sine and cosine functions to model real-life data.

Graph of Sine

2 2 2

2

2

32

3

1

1

Domain: All Real NumbersRange: [-1, 1]

Symmetry: Origin

Graph of Cosine

2 2 2

2

2

32

3

1

1

Domain: All Real NumbersRange: [-1, 1]

Symmetry: y-axis

Amplitude

The amplitude of the functions y = Asin and y = Acos is the absolute value of A.

The amplitude of a graph can be described as the absolute value of one-half the difference of the maximum and

minimum function values.

2 2 2

2

2

32

3

1

1

Example

State the amplitude of the function y = 3cos. Graph both y = 3cos and y = cos on the same coordinate plane.

A = 3

2 2 2

2

2

32

3

1

1

2

3

2

3

Period

The period of a function is the distance required to complete one full cycle.

kP

2

The period of the functions y = sink and y = cosk is given by:

2 2 2

2

2

32

3

1

1

Example

State the period of the function y = cos4. Graph both y = cos4 and y = cos on the same coordinate plane.

24

2 P

2 22

2

2

32

3

1

2

3

2

3

1

Amplitudes and Periods

14.2 Translations of Trig Graphs

The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C/B. The number C/B is called the phase shift.

amplitude = | A|

period = 2 /B.

The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C/B. The number C/B is called the phase shift.

amplitude = | A|

period = 2 /B.

x

y

Amplitude: | A|

Period: 2/B

y = A sin Bx

Starting point: x = C/B

The Graph of y = Asin(Bx - C)

Example

Determine the amplitude, period, and phase shift of y = 2sin(3x-)

Solution:

Amplitude = |A| = 2

period = 2/B = 2/3

phase shift = C/B = /3

Example cont.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3

-2

-1

1

2

3

• y = 2sin(3x- )

dcbxa )sin(Amplitude

Period:

2π/b Phase Shift:

c/b

Vertical

Shift

Phase Shift

A phase shift is simply a horizontal change.

The phase shift of the function y = A cosk( + C) is C. If C > 0, the shift is to the left. If C < 0, the shift is to the right.

2 2 2

2

2

32

3

1

1

Example

4cos

y

State the phase shift of the following function. Then sketch the function and y = cos on the same coordinate plane.

4

PS

2 2 2

2

2

32

3

1

1

2

3

2

3

Vertical Shift

A vertical shift is simply a vertical change.

The vertical shift of the function y = Asink + C is C. If C > 0, the shift is up. If C < 0, the shift is down.

2 2 2

2

2

32

3

1

1

Example

2cos y

State the vertical shift of the following function. Then sketch the function and y = cos on the same coordinate plane.

2VS

2 2 2

2

2

32

3

1

1

2

3

2

3

Practice

State the amplitude, period, phase shift, and vertical shift of the following function.

4cos6 y

4

2

6

VS

PS

P

A

Finding a Trigonometric Model

Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine, varies with the tides. The table below shows the depths (in feet) at various times during the morning. (a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon can it

safely dock?Time, Time, tt Depth, Depth, yy

MidnightMidnight 3.43.4

2 A.M.2 A.M. 8.78.7

4 A.M.4 A.M. 11.311.3

6 A.M.6 A.M. 9.19.1

8 A.M.8 A.M. 3.83.8

10 A.M.10 A.M. 0.10.1

NoonNoon 1.21.2

Begin by graphing the data. Let the time be the independent variable and the depth be the

dependent variable.

Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon

can it safely dock?

The model fits either a sine or

cosine model. Select cosine.

10

8

6

4

2

5 10

The amplitude is half the difference of the

max and min.a = 0.5(11.3 – 0.1)

a = 5.6

Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon

can it safely dock?

The period can be found by doubling

the difference of the low time and high

time.

10

8

6

4

2

5 10

p = 2(10 – 4)

p = 12

k

212

6

k

Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon

can it safely dock?

The high point should occur at the

origin. Thus there is a shift of 4 to the

right.

10

8

6

4

2

5 10

With an amplitude of 5.6, the high point has been raised by

5.7

Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon

can it safely dock? Given all these changes, the model

of best fit is10

8

6

4

2

5 10

7.546

cos6.5 ty

7.53

2

6cos6.5

ty

fty

fty

57.1015

84.09

The depth is at least 10 feet from 2:42 P.M. to

5:18 P.M.

HomeworkHomework14.1 pg 767 #’s 15, 18, 21, 24, 27, 30, 3314.1 pg 767 #’s 15, 18, 21, 24, 27, 30, 33

14.2 pg 774 #’s 19-24 all14.2 pg 774 #’s 19-24 all