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Section 5.2 - Graphs of the Sine and Cosine Functions
In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations. Much of what we will do in graphing these problems will be the same as earlier graphing using transformations.
Definition: A non-constant function f is said to be periodic if there is a number p > 0 such that f ( x + p ) = f ( ) x for all x in the domain of f. The smallest such number p is called the period of f.
The graphs of periodic functions display patterns that repeat themselves at regular intervals.
Definition: For a periodic function f with maximum value M and minimum value m.
The amplitude of the function is:. 2
M − m
In other words the amplitude is half the height.
Example 1: State the period and amplitude of the periodic function.
Example 2: State the period and amplitude of the periodic function:
Note: For a periodic function f , the period of the graph is the length of the interval needed to draw one
complete cycle of the graph. For a basic sine or cosine function, the period is 2 π.
For a basic sine or cosine function, the maximum value is 1 and the minimum value is -1, so the amplitude
is 1. 2
Drawing all of these points is rather tedious. We’ll ask you to learn the shape of the graph and just graph five basic points, the x and y intercepts and the maximum and the minimum.
Period: 2 π
Amplitude: 1
x-intercepts: π , 2π
y-intercept: (0,0)
Big picture: f ( x )=sin( x )
Period: 2 π
Amplitude: 1
x-intercepts: k π
y-intercept: (0,0)
Now we’ll repeat the process for the basic cosine function f ( x )= cos( x ). The domain of this function is
( − ∞, ∞)and the range is [-1, 1]. Again, we typically graph just one complete period of the graph, that is on
the interval [ 0 , 2 π].
Here is the table of values for f ( x )= cos( x ):
x 0
6
cos x
Now we’ll graph these ordered pairs.
Now we’ll turn our attention to transformations of the basic sine and cosine functions. These functions will be of the form f ( x )= A sin( Bx − C )+ D or g ( x )= A cos( Bx − C )+ D .We can stretch or shrink sine and
cosine functions, both vertically and horizontally. We can reflect them about the x axis, the y axis or both axes, and we can translate the graphs either vertically, horizontally or both. Next we’ll see how the values for A, B, C and D affect the graph of the sine or cosine function.
Graphing f^ ( ) x^^ =^ A^ sin(^ Bx^ −^ C^ )^ +^ D^ or^ g x ( )^^ =^ A^ cos(^ Bx^ −^ C^ )+ D
- The amplitude of the graph of is A.
- The period of the function is:.
B
- If A > 1, this will stretch the graph vertically. 0 < A < 1, this will shrink the graph vertically If A < 0, the graph will be a reflection about the x axis.
- If B >1 , this will shrink the graph horizontally by a factor of 1/B. If 0<B<1, this will stretch the graph horizontally by a factor of 1/B.
- Vertical Shift: Shift the original graph D units UP if D > 0, D units DOWN if D < 0.
- Phase shift: The function will be shifted B
C
units to the right if > 0 B
C
and to the left if < 0 B
C
. The
number B
C
is called the phase shift.
Note: Horizontal Shift: If the function is of the form f ( x )= sin( x − C )or f ( x )= cos( x − C ), then shift the original graph C units to the RIGHT if C > 0 and C units to the LEFT if C < 0.
Amplitude:
Transformations:
( ) cos 2
f x x
Period:
Amplitude:
Transformations:
f ( ) x = 2 cos ( x − π) Period:
Amplitude:
Transformations:
f ( ) x = −5sin 4 ( x + π) Period:
Amplitude:
Transformations:
f ( ) x = sin 2 ( x − π)+ 1 Period:
Amplitude:
Transformations:
( ) 5sin 1 2 8
x f x
π^ π
Period:
Amplitude
Transformations:
Example 5: Sketch over one period: ( ) cos 2
x f x
- Example 6: Sketch over one period: f ( ) x = 4 cos(2 π x ) −
Example 8: Sketch over one period:
( ) 2 cos 4 8
f x x
Exercise: Sketch over one period: ( ) 2sin 1 2 8
x f x
π^ π
Exercise: Consider the graph: Write an equation of the form f ( x )= A sin( Bx − C )+ D and an equation of
the form f ( x )= A cos( Bx − C )+ D which could be used to represent the graph. Note: these answers are not
unique!
Section 5.3a - Graphs of Secant and Cosecant Functions
Using the identity , sin( )
csc( ) x
x = you can conclude that the graph of g will have a vertical
asymptote whenever sin( x ) = 0. This means that the graph of g will have vertical asymptotes at
x = 0 ,± π ,± 2 π, …. The easiest way to draw a graph of g ( x )= csc( x )is to draw the graph of
f ( x )= sin( x ), sketch asymptotes at each of the zeros of f ( x )= sin( x ), then sketch in the
cosecant graph.
1 ( ) csc( ) sin( )
g x x x
= = ; if sin( ) x = 0 , then g(x) has a vertical asymptote.
Here’s the graph of f ( x )= sin( x )on the interval (^)
Next, we’ll include the asymptotes for the cosecant graph at each point where sin( x )= 0.
