Periodic Functions

Answer all periodic but d)

What are some examples of natural periodic functions in time?

Weather – monthly rain fall, temperature, seasons, Tides Amount of daylight Energy waves

Sine Graph:

Fill in the important points using exact angles and heights:

What is the period? Amplitude?

Cosine Graph:Fill in the important points using exact angles and heights:

What is the period? Amplitude?

What is the period? Amplitude? Vertical asymptotes?

What we should notice from the investigation is:

For : c affects horizontal translationd affects the vertical translation

For : a affects vertical stretch and |a| is the amplitude.

b affects the period. Period ¿2π|b|

General function:

Solutions:

State the transformations which map , and state the amplitude, period and principal axis for:

a) b)

c) d)

e) Answers:

a) Horizontal stretch of 1/2, vertical shift of -3amplitude = 1, period = π, principal axis of y = -3

b) Horizontal shift of −π2 , vertical shift of 1

amplitude = 1, period = 2π , principal axis of y = 1c) Horizontal shift of 2, vertical shift of 2, vertical stretch of 2

amplitude = 2, period = 2π , principal axis of y = 2

d) Horizontal stretch of 1/2, horizontal shift of −π4 ,vertical shift of -1

amplitude = 1, period = π, principal axis of y = -1

e) Horizontal stretch of π2 , horizontal shift of π, vertical stretch of

52 ,

vertical shift of -3, amplitude = 52 , period = π2, principal axis of y = -3