MAC 1114
Module 4
Graphs of the Circular
Functions
Rev.S08
Learning Objectives
•
Upon completing this module, you should be able to:
1.
2.
Recognize periodic functions.
Determine the amplitude and period, when given the equation
of a periodic function.
Find the phase shift and vertical shift, when given the
equation of a periodic function.
Graph sine and cosine functions.
Graph cosecant and secant functions.
Graph tangent and cotangent functions.
Interpret a trigonometric model.
3.
4.
5.
6.
7.
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Graphs of the Circular Functions
There are three major topics in this module:
- Graphs of the Sine and Cosine Functions
- Translations of the Graphs of the Sine and Cosine
Functions
- Graphs of the Other Circular Functions
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3
Introduction to Periodic Function

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 positive value of p is the period
of the function.
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4
Example of a Periodic Function
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Example of Another Periodic Function
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What is the Amplitude of a Periodic
Function?


The amplitude of a periodic function is half the
difference between the maximum and minimum
values.
The graph of y = a sin x or y = a cos x, with
a  0, will have the same shape as the graph of
y = sin x or y = cos x, respectively, except the
range will be [|a|, |a|]. The amplitude is |a|.
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How to Graph y = 3 sin(x) ?
x
0
/2

3/2

sin x
0
1
0
1
0
3sin x
0
3
0
3
0
Note the difference between sin x and 3sin x. What is the
difference?
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How to Graph y = sin(2x)?



The period is 2/2 = . The graph will complete
one period over the interval [0, ].
The endpoints are 0 and , the three middle
points are:
Plot points and join in a smooth curve.
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How to Graph y = sin(2x)?
(Cont.)
Note the difference between sin x and sin 2x. What is the
difference?
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Period of a Periodic Function
•Based on the previous example, we can
generalize the following:
For b > 0, the graph of y = sin bx will
resemble that of y = sin x, but with period 2/b.
The
graph of y = cos bx will resemble that of
y = cos x, with period 2/b.
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How to Graph y = cos (2x/3)
over one period?


The period is 3.
Divide the interval into four equal parts.


Obtain key points for one period.
x
0
3/4
3/2
9/4
3
2x/3
0
/2

3/2
2
cos 2x/3
1
0
1
0
1
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How to Graph y = cos(2x/3)
over one period? (Cont.)


The amplitude is 1.
Join the points and connect with a smooth curve.
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Guidelines for Sketching Graphs of Sine
and Cosine Functions




To graph y = a sin bx or y = a cos bx, with
b
> 0, follow these steps.
Step 1 Find the period, 2/b. Start with 0 on
the x-axis, and lay off a distance of
2/b.
Step 2 Divide the interval into four equal parts.
Step 3 Evaluate the function for each of the
five x-values resulting from Step 2.
The points will be maximum points,
minimum points, and x-intercepts.
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Guidelines for Sketching Graphs of Sine
and Cosine Functions Continued

Step 4

Step 5
Rev.S08
Plot the points found in Step 3, and join
them with a sinusoidal curve having
amplitude |a|.
Draw the graph over additional periods,
to the right and to the left, as needed.
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How to Graph y = 2 sin(4x)?

Step 1

Step 2
Period = 2/4 = /2. The function will
be graphed over the interval [0, /2] .
Divide the interval into four equal parts.

Step 3
Make a table of values
x
0
/8
/4
3/8
/2
4x
0
/2

3/2
2
sin 4x
0
1
0
1
0
2 sin 4x
0
2
0
2
0
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How to Graph y = 2 sin(4x)? (Cont.)

Plot the points and join them with a sinusoidal
curve with amplitude 2.
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What is a Phase Shift?

In trigonometric functions, a horizontal translation
is called a phase shift.

In the equation
the graph is shifted /2 units to the right.
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How to Graph y = sin (x  /3) by Using
Horizontal Translation or Phase Shift?

Find the interval for one period.

Divide the interval into four equal parts.
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How to Graph y = sin (x  /3) by Using
Horizontal Translation or Phase Shift?
(Cont.)
x
/3 5/6 4/3
11/6
7/3
x  /3
0
/2

3/2
2
sin (x  /3)
0
1
0
1
0
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How to Graph y = 3 cos(x + /4) by Using
Horizontal Translation or Phase Shift?

Find the interval.

Divide into four equal parts.
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How to Graph y = 3 cos(x + /4) by Using
Horizontal Translation or Phase Shift?
x
x + /4
cos(x + /4)
3 cos (x + /4)
Rev.S08
/4
0
1
3
/4
/2
0
0
3/4

1
3
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5/4
3/2
0
0
7/4
2
1
3
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How to Graph y = 2  2 sin 3x by Using
Vertical Translation or Vertical Shift?

The graph is translated 2 units up from the graph
y = 2 sin 3x.
x
0
/6
/3
/2
2/3
3x
0
/2

3/2
2
2 sin 3x
0
2
0
2
0
2  2 sin 3x
2
0
2
4
2
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How to Graph y = 2  2 sin 3x by Using
Vertical Translation or Vertical Shift?
(Cont.)

Plot the points and connect.
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Further Guidelines for Sketching Graphs
of Sine and Cosine Functions
Method 1: Follow these steps.
Step 1
Find an interval whose length is one
period 2/b by solving the three part
inequality 0  b(x  d)  2.
Step 2
Divide the interval into four equal parts.
Step 3
Evaluate the function for each of the
five x-values resulting from Step 2. The
points will be maximum points,
minimum points, and points that intersect the
line y = c (middle points of the wave.)
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Further Guidelines for Sketching Graphs
of Sine and Cosine Functions (Cont.)

Step 4

Step 5
Method 2:
Rev.S08
Plot the points found in Step 3, and join
them with a sinusoidal curve having
amplitude |a|.
Draw the graph over additional periods,
to the right and to the left, as needed.
First graph the basic circular function. The
amplitude of the function is |a|, and the period
is 2/b. Then use translations to graph the
desired function. The vertical translation is c
units up if c > 0 and |c| units down if c < 0.
The horizontal translation (phase shift) is d
units to the right if d > 0 and |d| units to the
left if d < 0.
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How to Graph y = 1 + 2 sin (4x + )?

Write the expression in the
form c + a sin b(x  d) by
rewriting 4x +  as

Step 1
Rev.S08

Step 2: Divide the
interval.

Step 3 Table
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How to Graph y = 1 + 2 sin (4x + )?(Cont.)
/4
/8
0
/8
/4
x + /4
0
/8
/4
3/8
/2
4(x + /4)
0
/2

3/2
2
sin 4(x + /4)
0
1
0
1
0
2 sin 4(x + /4)
0
2
0
2
2
1
1
1
3
1
x
1 + 2sin(4x + )
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How to Graph y = 1 + 2 sin (4x + )?
(Cont.)

Steps 4 and 5

Plot the points found in the table and join then with a
sinusoidal curve.
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Let’s Take a Look at Other Circular
Functions.
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Cosecant Function
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Secant Function
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Guidelines for Sketching Graphs of
Cosecant and Secant Functions


To graph y = csc bx or y = sec bx, with b > 0,
follow these steps.
Step 1 Graph the corresponding reciprocal
function as a guide, using a dashed
curve.
To Graph
Rev.S08
Use as a Guide
y = a csc bx
y = a sin bx
y = a sec bx
y = cos bx
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Guidelines for Sketching Graphs of
Cosecant and Secant Functions Continued

Step 2
Sketch the vertical asymptotes.
- They will have equations of the form x = k,
where k is an x-intercept of the graph
of the guide function.

Step 3
Sketch the graph of the desired function
by drawing the typical U-shapes branches
between the adjacent asymptotes.
- The branches will be above the graph of the
guide function when the guide function values
are positive and below the graph of the guide
function when the guide function values are
negative.
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How to Graph y = 2 sec(x/2)?
Step 1: Graph the corresponding reciprocal function
y = 2 cos (x/2).
The function has amplitude 2 and one period of the graph
lies along the interval that satisfies the inequality
Divide the interval into four equal parts.
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How to Graph y = 2 sec(x/2)? (Cont.)


Step 2: Sketch the vertical asymptotes. These occur at xvalues for which the guide function equals 0, such as x =
3, x = 3, x = , x = 3.
Step 3: Sketch the graph of y = 2 sec x/2 by drawing the
typical U-shaped branches, approaching the asymptotes.
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Tangent Function
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Cotangent Function
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Guidelines for Sketching Graphs of
Tangent and Cotangent Functions


To graph y = tan bx or y = cot bx, with b > 0,
follow these steps.
Step 1 Determine the period, /b. To locate
two adjacent vertical asymptotes solve
the following equations for x:
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Guidelines for Sketching Graphs of Tangent
and Cotangent Functions continued

Step 2

Step 3

Step 4

Step 5
Rev.S08
Sketch the two vertical asymptotes found in
Step 1.
Divide the interval formed by the vertical
asymptotes into four equal parts.
Evaluate the function for the first-quarter
point, midpoint, and third-quarter point, using
the x-values found in Step 3.
Join the points with a smooth curve,
approaching the vertical asymptotes. Indicate
additional asymptotes and periods of the
graph as necessary.
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How to Graph y = tan(2x)?





The period of the function is /2. The
two asymptotes have equations
x = /4 and x = /4.
Step 2:
Sketch the two vertical asymptotes
found.
x =  /4.
Step 3:
Divide the interval into four equal parts.
This gives the following key x-values.
First quarter: /8
Middle value: 0
Third quarter: /8
Step 1:
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How to Graph y = tan(2x)? (Cont.)

Step 4:
x
/8
0
/8
2x
/4
0
/4
1
0
1
tan 2x

Evaluate the function
Step 5:
Join the points with a
smooth curve, approaching the
vertical asymptotes. Indicate
additional asymptotes and periods of
the graph as necessary.
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How to Graph y = tan(2x)? (Cont.)

Every y value for this
function will be 2 units
more than the
corresponding y in
y = tan x, causing the
graph to be translated 2
units up compared to
y = tan x.
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What have we learned?
•
We have learned to:
1.
2.
Recognize periodic functions.
Determine the amplitude and period, when given the equation
of a periodic function.
Find the phase shift and vertical shift, when given the
equation of a periodic function.
Graph sine and cosine functions.
Graph cosecant and secant functions.
Graph tangent and cotangent functions.
Interpret a trigonometric model.
3.
4.
5.
6.
7.
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Credit
•
Some of these slides have been adapted/modified in part/whole
from the slides of the following textbook:
•
Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th
Edition
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