Download Graphing Sine and Cosine Functions

LESSON 6: GRAPHING SINE AND COSINE FUNCTIONS
Learning Outcome:




To sketch the graphs of y = sin x and y = cos x
To determine the characteristics of the graphs of y = sin x and y = cos x
To demonstrate an understanding of the effects of vertical and horizontal stretches on the
graphs of sinusoidal functions
To solve a problems by analyzing the graph of a trigonometric function
Periodic Functions:
A periodic function is a function whose graph repeats regularly over some interval of the
domain. The length of this interval is called the period of the function.
The amplitude of a periodic function is defined as half the distance between the maximum and
minimum values of the function.
max  min
amplitude 
2
Graph y  sin  and y  cos  for 0    2
Sin x
Cos x


Must be in radians mode: 0, 4 ,   2, 2,1
2

Graph y  2sin  for 0    2
amplitude
2
1

-1
2

3
2
-2
Period
2
For y  a sin 
As a increases or
decreases, the
period stays the
same and the
amplitude increases
or decreases.
a is a vertical
stretch.
Ex. Graph: y=sin x, y=3sin x, and y=0.5sin x, for 0    2 . State the amplitude of each.
Ex. Sketch y=cos x and y=cos 2x, for 0    360
Ex. Now graph y=cos 3x and y=cos0.5x over the same interval. What did you notice?
In general:
To find the period of a function in the form y=a sin bx or y=a cos bx:
period =
2
360
(for degrees) period =
(for radians)
b
b
Ex. State the amplitude and period of each:
a. y  2 cos 3 :
b. y 
1
1
cos  :
2
3
Now we will consider the graphs of the functions whose equations are;
y  a sin b  x  c    d
and
y  a cos b  x  c    d
What transformations are occurring in the following examples:
a.
b.
c.
d.
y= 2 sin x :
y=sin 2x:
y=-3sin x:
y=sin (-3x):
Notice: vertical stretches affect amplitude, and horizontal stretches affect period.
Ex. a. List the steps involved in graphing the function y = 3 sin 4x.
b. Determine:

Amplitude

Period

Maximum and minimum value

x-intercept, y-intercept

Domain and range
One cycle of sin x has the following characteristics:
Max value of 1, min value of -1, amplitude of 1, period of 2π, y-intercept of 0, x-intercepts of 0, π
and 2π, domain of 𝜃𝜖𝑅, and a range of −1 ≤ 𝑦 ≤ 1
How many of the parameters above will change if we compare to the graph of y = cos x
Max/min value:
Amplitude/period:
Domain/range:
y-intercept
x-intercepts
Assignment: pg.233-237 #1-12, 14, 15, 19, 20, 21, 23