4.1 – Basic Graphs
The Sine Graph
x
y = sin x
0
4
2
3
4
5
4
3
2
7
4
2
*Consider the unit circle definition of sine:
y = sin t
At (1, 0 ) , sin t = 0 .
As a point P moves along the circumference of the unit
circle toward ( 0,1) or t = 2 , sin t increases from 0 to 1.
As a point P moves along the circumference of the unit
circle from ( 0,1) to ( 1, 0 ) , that is from t = 2 to t = ,
sin t decreases from 1 to 0.
This relationship continues to trace out the entire sine
graph.
*Above we have one complete cycle of y = sin x . If we go beyond 2 , t will take on
values that are coterminal with the angles between 0 and 2 .
Math 124 Ch. 4
McKeague 8th Ed.
Definition: Period
For any function y = f ( x ) , the smallest positive number p for which
f ( x + p ) = f ( x ) , for all x
is called the period of f ( x ) .
inverse
a function
is found by interchanging the coordinates in
What is the The
period
of theofsine
function?
each ordered pair that satisfies the function.
Definition: Amplitude
If the greatest value of y is M and the least value of y is m , then
the amplitude of the graph of y is defined as
A=
1
2
M
m.
What is the amplitude of the sine function?
Definition: Zero
A zero of a function y = f ( x ) is any domain value x = c for which
f ( c ) = 0 . If c is a real number then x = c is an x-intercept of the graph
of f .
What are the zeros of the sine function?
What are the domain and range of the sine function?
Math 124 Ch. 4
McKeague 8th Ed.
The Cosine Graph
x
y = cos x
0
4
2
3
4
5
4
3
2
7
4
2
The Tangent Graph
x
y = sin x
3
4
6
0
6
4
3
2
2
3
3
4
Math 124 Ch. 4
McKeague 8th Ed.
The Graphs of Cosecant, Secant, and Cotangent
Even and Odd Functions
Definition: Even Function
An even function is a function for which
f ( x) = f ( x)
for all x in the domain of f .
*The graph of an even function is symmetric about the y-axis.
Definition: Odd Function
An odd function is a function for which
f ( x) = f ( x)
for all x in the domain of f .
*The graph of an odd function is symmetric about the origin.
Math 124 Ch. 4
McKeague 8th Ed.
o What about sine and cosine?
Example 1: Show that secant is an even function.
Example 2: Use the even and odd function relationships to find the exact value of each of
a. sin ( 30 )
b. cos ( 30
)
c. cos ( 150
)
Math 124 Ch. 4
McKeague 8th Ed.
4.2 – Amplitude, Reflection, and Period
Amplitude – vertical stretch/compression
Example 1: Sketch the graph of y = 2cos x for 0
x
x
2 .
y = 2cos x
Example 2: Sketch the graph of y = 12 sin x for 0 £ x £ 2p .
x
y = 12 sin x
*If A 0 , then the graphs of y = Asin x and y = Acos x will have amplitude A and
range A, A .
Math 124 Ch. 4
McKeague 8th Ed.
Reflection about x-axis
Example 3: Sketch the graph of y = 2sin x from x = 2
to x = 4 .
*The graphs of y = Asin x and y = Acos x will be reflected about the x-axis if A 0 . The
amplitude will be A .
Period
Example 4: Sketch the graph of y = cos 2 x for 0 £ x £ 2p . Use a calculator to build a table of
values.
So the graph of y = cos 2 x has period: __________
*In general, notice that cosine and sine each complete one cycle when the argument varies from
0 to 2 . That is:
0 argument 2
Math 124 Ch. 4
McKeague 8th Ed.
Example 5: Graph one complete cycle of
y = sin 12 x .
*In general, for y = sin Bx and y = cos Bx to complete one cycle, we have
2
0 Bx 2
0 x
or
.
B
So the period is 2B and the graph completes B cycles in 2 units.
Amplitude and Period for Sine and Cosine
If A is any real number and B 0 , then the graphs of y = A sin Bx and y = A cos Bx will have
2
Amplitude = A
Period =
and
B
Example 6: Graph y = 3cos (
2
3
x ) for
15
4
x
15
4
.
Math 124 Ch. 4
McKeague 8th Ed.
4.3 – Vertical and Horizontal Translations
Vertical Tanslation
*Recall from algebra:
Example 1: Sketch the graph of y = 3 + 2 cos ( 12 x ) for 4
x
4 .
*In general, the graphs of y = k + sin x and y = k + cos x will be translated vertically k units
upward if k 0 and k units downward if k 0 .
Math 124 Ch. 4
McKeague 8th Ed.
Horizontal Translation
Example 2: Graph y = sin ( x +
2
) for
2
x
3
2
.
x
*So the graph of y = sin ( x +
) is shifted
y = sin ( x +
2
)
units to the left of the graph of y = sin x .
How would we do example 2 without a table?
2
2
Example 3: Graph on complete cycle of y = cos ( x
4
).
Math 124 Ch. 4
McKeague 8th Ed.
*The graphs of y = sin ( x h ) and y = cos ( x h ) will be translated horizontally h units to:
the right if h 0
the left if h 0
The value h is called the horizontal translation or phase shift.
Example 4: Graph one complete cycle of y = 3cos ( 12 x +
3
).
Graphing the Sine and Cosine Functions
The graphs of y = k + A sin ( B ( x h ) ) and y = k + A cos ( B ( x h ) ) where B
0 , will
have:
Amplitude = A
Horizontal Translation = h
If A 0 , the graph will be reflected about the x-axis.
2
B
Vertical Translation = k
Period =
Example 5: Graph one complete cycle of y = 3 2sin ( x +
2
).
Math 124 Ch. 4
McKeague 8th Ed.