Graphing Sine and Cosine Functions - MRS. POWER...Pre-Calculus 12A Sections 5.1/5.2 3 THE SINE...

Pre-Calculus 12A Sections 5.1/5.2

1

Graphing Sine and Cosine Functions

PERIODIC FUNCTION:

a function for which the dependent variable takes on the same set of values

over and over again as the independent variable changes.

Examples:

The automatic dishwasher in a school cafeteria runs constantly

through lunch. The graph shows the amount of water used as a function of time.

An individual’s height above the ground as a

function of time as they ride the ferris wheel.

An ECG measures the electrical activity of a

person’s heart in millivolts over time.

Rebecca has a submersible pump in her basement

that is situated such that it can remove water that

collects below her foundation. There is a hole in the

basement floor and the pump sits in the bottom of

that hole. During a particularly heavy rain storm, the

pump kept turning off and on at regular intervals as

it attempted to drain the excess water below the

foundation. The relationship between the depth of

water in the hole and time is shown in the graph.


2

Determine whether each function is or is not periodic.

CYCLE:

the smallest portion of the graph that contains the complete repeating pattern.

PERIOD:

the horizontal distance required for the graph of a periodic function to

complete one cycle.

Examples of periodic functions:

distance pendulum swings from its rest position

tide levels

clocks

roller coasters

traffic lights


3

THE SINE FUNCTION

Sketch the graph of siny for 0 360 or 0 2 . Describe its characteristics.

Solution: siny

Complete the following table of values . Plot the points and join them with a smooth curve.

0

30

45

60

90

120

135

150

180

210

225

240

270

300

315

330

360

0

6

4

3

2

2

3

3

4

5

6

7

6

5

4

4

3

3

2

5

3

7

4

11

6

2

sin

exact

2

2

sinapp

0.71


4

EQUATION OF THE SINUSOIDAL AXIS:

the equation of the horizontal line halfway between the maximum and minimum.

AMPLITUDE:

the maximum vertical distance from the function’s axis to the maximum or minimum value.

Using your graph of the sine function, determine each of the following:

DOMAIN:

RANGE:

MAXIMUM VALUE:

MINIMUM VALUE:

EQUATION OF THE

SINUSOIDAL AXIS:

AMPLITUDE:

PERIOD:

Y-INTERCEPT

-INTERCEPTS

Determine the 5 key points that would enable you to draw the sine curve quickly, in both degrees and radians.

(degrees) y

0

90

180

270

360

SINUSOIDAL FUNCTION:

a periodic function whose graph looks like smooth symmetrical waves,

where any portion of the wave can be horizontally translated onto

another portion of the curve.

(radians) y

0

2

3

2

2


5

TRANSFORMATIONS OF THE SINE FUNCTION

Sketch the graph of the following functions over the given intervals. Identify the amplitude, period, vertical

translation, phase shift, domain, range, maximum and minimum values, and the equation of the sinusoidal axis.

a. 2sin( 45 ) 3y x for the interval 360 ,360

b. 3sin2

4

y x

for the interval 2 ,2

Solution:

a. Mapping Rule: ( , )x y ________________________

siny 2sin( 45 ) 3y x

(degrees) y

0

90

180

270

360

(degrees) y

2sin( 45 ) 3y x

Amplitude:

Max Value - Min ValueAmplitude =

2

Period:

Vertical Translation:

Phase Shift (Horizontal

Translation):

Domain:

Range:

Maximum Value:

Minimum Value:

Equation of the sinusoidal axis:


6

b. Mapping Rule: ( , )x y ________________________

siny 3sin2

4

y x

(radians) y

(radians) y

3sin2

4

y x

Amplitude:

Period:


Phase Shift:

Domain:

Range:

Maximum Value:

Minimum Value:

Equation of the Sinusoidal Axis:

When the equation of a sinusoidal function is given in the form

sin ( )

cos ( )

y a b x c d

y a b x c d

, then

Vertical stretch is by a factor of a . Thus the amplitude is a . If 0a , the function is reflected in the x-

axis.

Horizontal stretch is by a factor of 1

b

. Thus the period is changed to 360 2

or

bb

. Reflected in the y-axis if

b<0.

Vertical displacement is represented by d, up if d > 0 and down if d < 0.

Horizontal phase shift is represented by c, to the right if c > 0, and to the left if c < 0.


7

DETERMINE AN EQUATION FROM A GRAPH

Each of the graphs below illustrates the function of ( )y f x . Write the equation of the function in the

form sin ( )y a b x c d .

a.

b.

SOLUTION A:

Amplitude:

Period:

Equation of Sinusoidal Axis:

Phase Shift:

Vertical Stretch:

Horizontal Stretch:


Horizontal Translation:

EQUATION:

SOLUTION B:

Amplitude: Period: Equation of the Sinusoidal Axis: Phase Shift:

Vertical Stretch : Horizontal Stretch: Vertical Translation: Horizontal Translation:

EQUATION: ________________________________________________


8

THE COSINE FUNCTION

Sketch the graph of cosy for 0 360 or 0 2 . Describe its characteristics.

Solution:

Complete the following table of values. Plot the points and join them with a smooth curve.

0

30

45

60

90

120

135

150

180

210

225

240

270

300

315

330

360

0

6

4

3

2

2

3

3

4

5

6

7

6

5

4

4

3

3

2

5

3

7

4

11

6

2

cos

exact

2

2

cosapp

0.71

Determine the 5 key points that would enable you to draw the cosine curve quickly, in both degrees and

radians. Write your solutions in the tables provided below.

Using your graph of the cosine function, determine each of the following:

DOMAIN:

RANGE:

MAXIMUM VALUE:

MINIMUM VALUE:

EQUATION OF THE

SINUSOIDAL AXIS:

AMPLITUDE:

PERIOD:

Y-INTERCEPT:

-INTERCEPTS :

(degrees) cosy

0

90

180

270

360

(radians) cosy

0

2

3

2

2


9

TRANSFORMATIONS OF THE COSINE FUNCTION

Sketch the graph of the following functions over the given interval. Identify the amplitude, period, vertical

translation, phase shift, domain, range, maximum and minimum values, and the equation of the

sinusoidal axis.

a. 1

3cos 60 1

4

y x

over the interval 120 ,1740

b. 1cos 2

4 6

y

over the interval ,

-2 11

3 6

Solution:

a. Mapping Rule: ___________________________

y=cosine 1

3cos 60 1

4

y x

(degrees) y

0

90

180

270

360

(degrees)

y

1

3cos 60 1

4

y x

Amplitude:

Period:


Phase Shift:

Domain:

Range:

Maximum Value:

Minimum Value:

Equation of the Sinusoidal Axis:


10

b. Mapping Rule: ___________________________

cosy 1cos 2

4 6

y

(radians) y

(radians) y

1cos 2

4 6

y

Amplitude:

Period:


Phase Shift:

Domain:

Range:

Maximum Value:

Minimum Value:

Equation of the Sinusoidal Axis;


11

x

y

DETERMINE AN EQUATION FROM A GRAPH

Each of the graphs below illustrates the function of ( )y f x . Write the equation of the function in the form

a. cos ( )y a b x c d

b. sin ( )y a b x c d

SOLUTION A:

Amplitude:

Period:


Phase Shift

Vertical Stretch

Horizontal Stretch



EQUATION 1:

EQUATION 2:

Solution B

Amplitude: Period:

Equation of Sinusoidal Axis

Phase Shift:

Vertical Stretch Horizontal Shift


Horizontal Translation

EQUATION 1

EQUATION 2


12

EXTRA PRACTICE

Each of the graphs below illustrates the function of ( )y f x . Write the equation of the function in the form

a. cos ( )y a b x c d

b. sin ( )y a b x c d

SOLUTION A:

Amplitude:

Period:


Phase Shift:

Vertical Stretch:

Horizontal Stretch:



EQUATION 1:

EQUATION 2:

Solution B

Amplitude: Period:


Phase Shift:

Vertical Stretch: Horizontal Stretch:



EQUATION 1

EQUATION 2

Graphing Sine and Cosine Functions - MRS. POWER...Pre-Calculus 12A Sections 5.1/5.2 3 THE SINE...

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