Graphing Form of Sine and Cosine Functions

Period

The length of one cycle of a graph.

Initial Trigonometric Graphing Form

siny a x h k Sine

Cosine

cosy a x h k

Do not write these

on your worksheet yet. We still need

to add one more

parameter.

Requirements for a Sine/Cosine Graph

x-intercept

At least one Period

(in other words, at least 5 consecutive critical points accurately plotted)

1

2

3

4

5

Arrows (to show that there

infinite cycles)

a > 10<a<1a < 0Amplitude:

Half of the distance between the maximum and minimum values of the range of a periodic function with a bounded

range.

The amplitude is the absolute value of a! It is a positive distance.

The Amplitude and the Effect of “a”

a = 1

1Amplitude =

3 0.5 1 0.5siny x

3siny x

siny x siny x

Example: Sine

32siny x y = 0

x = -3π/2

Transformation: Flip the parent graph and translate it 3Pi/2 units to the left.

Transformation:

22

New Equation:

Period:

2You need at least 5 consecutive critical points.

Example: Cosine

2cos 1y x

y = -1

x = -π/2

Transformation: Translate the parent graph Pi/2 units to the left and 1 unit down.

Transformation:

22

New Equation:

Period:

2You need at least 5 consecutive critical points.

Cosine

Sine v Cosine

Sine

(Press the Graph)

Translation - 3 units up and …

Example: Sine or Cosine?

y = 3

New Equation:

Transformation:

22

Amplitude - 2

Orientation -

Graph -

Period - 2π

Since the Sine and Cosine graphs are periodic and

translations of each other, there are infinite equations that

represent the same curve. Here are two examples.

3 units up and 3π/4 to the left

Example: Sine or Cosine?

342sin 3y x

x = -3π/4

New Equation:

Translation - 3 units up and … Transformation: Amplitude -

Orientation -

2 Graph - Sine

Positive

22

y = 3

Period - 2π

OR

3 units up and π/4 to the left

Example: Sine or Cosine?

42cos 3y x

x = -π/4

New Equation:

Translation - 3 units up and … Transformation: Amplitude -

Orientation -

2 Graph - Cosine

Positive

22

y = 3

Period - 2π

Changing the PeriodFind the period for each graph and generalize the result.

siny x 14cosy x

cos 4y x sin 2y x

1 cycle in 2π

2 2

22

Period = 2π

1/4 cycle in 2π

Period = 8π

2 cycles in 2π

Period = π

4 cycles in 2π

Period = 0.5π

2The coefficient of Period x

21 Period 2 2

1 4 Period 2 4 8

22 Period 2 1

4 2Period

Determining the Period of Sine/Cosine Graph

If or , the period (the length of one cycle) is determined by:

Ex: What is the period of ?

siny bx cosy bx

2 Period b

7sin 3 2f x x

23 Period

Changing the Period w/o Affecting (h,k)

sin 2 3 4y x

sin 2 3 4y x or

The key point (h,k) is a point on the sine graph. Also, multiplying x by a constant changes the period. Below are two different ways to write a transformation. In order for the equation to be useful, it must directly change the graph in a specific manner. Which equation changes the period and contains the point (-3,4)?

Graphing Form for Sine

siny a b x h k

h

k

Period 2 b

a

k a

k a

Amplitude:

Period: 2b

a

Graphing Form for Cosine

cosy a b x h k

h

k

Period 2 b

a

k a

k a

Amplitude:

Period: 2b

a

Notation: Trigonometric Functions

sin 2 56

y x

sin 2 56

y x

is equivalent toCorrect way for

the calculator!

Example: Sine

0.5sin 2 1y x

y = -1

x = π/2

Transformation: Change the amplitude to 0.5 and the period to π. Then translate it π/2 units

to the right and 1 unit down.

New Equation:

Transformation:

22

You need at least 5 consecutive critical points.

22 Period:

20.5sin 2 1y x

Not in Graphing

form

Example: Cosine

12cos 1y x

y = 1

x = 0

Transformation: Change the period to 4π and translate the parent graph 1 unit up.

Transformation:

22

You need at least 5 consecutive critical points.

New Equation:

21 2 Period:

4

Translation - 2 units down and …

Example: Sine or Cosine?

y = -2

New Equation:

Transformation:

2

Amplitude - 1.5

Orientation -

Graph -

Period - π/2

22 b

4b

Period:

Since the Sine and Cosine graphs are periodic and translations of each other, there are infinite equations

that represent the same curve. Here are two examples.

2 units down

Example: Sine or Cosine?

1.5cos 4 2y x

x = 0

New Equation:

Translation - Transformation: Amplitude -

Orientation -

Graph - Cosine

Positive

1.5 2 units down and …

22 b

4b

Period:

Period - π/2

2

y = -2

OR

2 units down and 5π/8 to the right

Example: Sine or Cosine?

3 52 8sin 4 2y x

x = 5π/8

New Equation:

Translation - Transformation: Amplitude -

Orientation -

Graph - Sine

Negative

1.5 2 units down and …

22 b

4b

Period:

Period - π/2

2

y = -2