6-3 6-4 Graphing Sine and Cosine

Periodic function: A function which has a graph that repeats itself identically over and over as it is followed from left to right. Formally, a function f is periodic if there exists a number p such that f(x + p) = f(x) for all x.

Period: the horizontal distance required to complete a full ‘cycle’

Examples:1) Find the period of each graph

Graph of y =sinx

a) What is the maximum values? When does it occur?

b) What is the minimum value? When does it occur?

c) What is the period?

d) What is the y intercept?

e) What are the x intercepts? How many are there?

Facts about Sine Graph:

Graph of y =cosx

a) What is the maximum values? When does it occur?

b) What is the minimum value? When does it occur?

c) What is the period?

d) What is the y intercept?

e) What are the x intercepts? How many are there?

Facts about Cosine Graph:

examples

examples

Use the graph of y = sinx to find the following values

4) –π/2

5)

6) -3π

7) Graph y=cos(x) for 3π ≤ x ≤5π

Amplitude: half the distance between the minimum and maximum values of the range of a periodic function. (the ‘height’ of the function

The amplitude of y=Asin(θ) or y=A cos(θ) is IAI

The period of y= sin kθ or y =cos kθ is

For example, given the equation y= 3 sin6θ,

The amplitude is 3 and the period is or

8) Find the amplitude of the graph y = 4cos (x)

9) Graph y=4cos(x) and y=cos(x) and then compare.

10)Find the amplitude and period for the function y = -2cos(4θ)

11) Write a sine function with an amplitude of 6.2 and period of 6π