Chapter 4
Graphs of the Circular Function
Section 4.1 Graphs of Sine and Cosine
Section 4.2 Translations of Sin and Cos
Section 4.3 Other Circular Functions
Section 4.1 Graphs of Sin & Cos
•
•
•
•
•
Identify Periodic Functions
Graph the Sine Function
Graph the Cosine Function
Identify Amplitude and Period
Use a Trigonometric Model
Periodic Functions
A periodic function is a function f
such that:
f(x) = f(x + np)
for every real number x in the
domain of f, every integer n, and
some positive real number p.
The smallest possible value of p
is the period of the function.
Graph of the Sine Function
POSEIDON/TOPEX Imagery
Graph of the Sine Function
Characteristics of the Sine Function.
Domain: (-ë, ë)
Range: [-1, 1]
Over the interval [0, é/2] 0 æ 1
Over the interval [é/2, é] 1 æ 0
Over the interval [é, 3é/2] 0 æ -1
Over the interval [3é/2, 2é] -1æ 0
The graph is continuous over its
entire domain and symmetric with
repeat to the origin.
x-intercepts:né
Period: 2é
Graph of the Cosine Function
Characteristics of the Cosine Function.
Domain: (-ë, ë)
Range: [-1, 1]
Over the interval [0, é/2] 1 æ 0
Over the interval [é/2, é] 0 æ-1
Over the interval [é, 3é/2] -1æ 0
Over the interval [3é/2, 2é] 0æ 1
The graph is continuous over its entire
domain and symmetric with repeat to
the origin.
x-intercepts: é/2 + né
Period: 2é
Amplitude of Sine and Cosine Functions
Example with a sound wave
The graph of
y= a sin x
or
y = a cos x, with a å 0,
will have the same shape as the
graph of
y = sin x
or
y= cos x,
respectively, except with the range
[-|a|, |a|].
|a| is called the amplitude.
Period of Sine and Cosine Functions
For b> 0, the graph of y = sin bx will look
like that of y = sin x, but with a period
of 2é/b.
Also the graph of y = cos bx will look like
that of y = cos x, but with a period of
2é/b.
Guidelines for Sketching
Graphs of Sine and Cosine
1. Find the period
2. Divide the interval into four equal parts
3. Evaluate the function for each of the five
x-values resulting from step 2.
4. Plot the points and join them with a
sinusoidal curve.
5. Draw additional cycles on the right and
left as needed.
Section 4.2 Translations of the
Graphs of Sin and Cos
•
•
•
•
Understand Horizontal Translations
Understand Vertical Translations
Understand Combinations of Translations
Determine a Trigonometric Maodel using
Curve Fitting
Horizontal Translations
• A horizontal translation is called a phase
shift when dealing with circular functions.
In the function y = f(x-d), the expression
(x-d) is called the argument with a shift of
d units to the right if d >0 and |d| units to
the left if d<0.
Vertical Translations
The graph of a function of the form
y = c + f(x)
is translated vertically as compared to the
graph of y = f(x) with a shift of c units up if
c >0 and |c| units down if c<0.
Combinations of Translations
The graph of a function of the form
y = c + f(x - d)
has both a horizontal and a vertical shift.
To graph the function it doesn’t matter
which one you look at first.
Determining a Trig Model
Using Curve Fitting
• http://mathdemos.gcsu.edu/mathdemos/si
nusoidapp/sinusoidapp.html
Section 4.3 Graphs of the Other
Circular Functions
•
•
•
•
•
Graph the Cosecant
Graph the Secant
Graph the Tangent
Graph the Cotangent
Understand Addition of Ordinates
Sine Graph
Cosine Graph
Cosecant Graph
Secant Graph
Tangent Graph
Cotangent Graph
Addition of Ordinates
• New functions can be formed by combining
other functions.
Example:
y = sin x + cos x
• Since the y coordinate is called the ordinate
Addition of ordinates means we add to get
the y coordinate
(x, sin x + cos x)
• On the graphing calculator we use Y1= sin x
and Y2= cos x with Y3= Y1 + Y2