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Section 5.2: Graphs of the Sine and Cosine Functions
A Periodic Function and Its Period
A nonconstant function f is said to be periodic if there is a number p > 0 such that
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.
Amplitude
Let f be a periodic function and let m and M denote, respectively, the minimum and maximum
values of the function. Then the amplitude of f is the number 2
M m
Example 1: Specify the period and amplitude of the given function
Now let’s talk about the graphs of the sine and cosine functions.
Recall: and
This means that after going around the unit circle once ( 2 π radians), both functions repeat. So
the period of both sine and cosine is 2 π. Hence, we can find the whole number line wrapped
around the unit circle.
Since the period of the sine function is 2π, we will graph the function on the interval [0, 2π], since
the rest of the graph will repeat itself.
Let’s take a look at Sine
Example 2
Sine : f (x )sinx
The big picture:
Since the period of the cosine function is 2π, we will graph the function on the interval [0, 2π], since
the rest of the graph will repeat itself.
So let’s take a look at the Cosine function.
2
3
2
Domain:____________________
Range:__________________
Period:__________________
Amplitude:_________________
x - intercepts:______________
y - intercept:_______________
Using the fact that cos( ) 2
sin
. These graphs will be translations, reflections, “stretches”,
and “squishes” of and_._
For the following functions:
y Asin(BxC) and y Acos(BxC)
Amplitude = A (Note: Amplitude is always positive.) If A is negative, that means an x - axis
reflection.
Period =
B
Translation in horizontal direction (called the phase shift ) =
B
C
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.
One complete cycle of the sine curve includes three x -intercepts, one maximum point and one minimum
point. The graph has x -intercepts at the beginning, middle, and end of its full period.
One complete cycle of the cosine curve includes two x -intercepts, two maximum points and one
minimum point. The graph has x -intercepts at the second and fourth points of its full period.
Key points in graphing these functions are obtained by dividing the period into four equal parts.
Example 4: Give the amplitude, period, and phase shift for the following functions:
a.
( ) 2 cos
f x x
b.
( ) 3 sin
f x x
c.
( ) sin
f x x
Example 5: Sketch over one period: f(x) 4 cos( 2 x)
Example 8: Give a function of the form and
which could be used to represent the graph. Note: these answers are not unique.
Example 9: Give a function of the form and
which could be used to represent the graph. Note: these answers are not unique.
