Graphs of
Trigonometric
Functions
Symmetry with respect to the
axis or line
A graph is said to be symmetric
with respect to a line if the
reflection (mirror image) about
the line of every point on the
graph is also on the graph The
line is known as the line of
symmetry.
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Symmetry with respect to a
point
A graph is said to be
symmetric with respect to a
point Q if to each point P on
the graph, we can find point P’
on the same graph, such that Q
is the midpoint of the segment
joining P and P’.
2
Two points are symmetric with respect to the y – axis
if and only if their x – coordinates are additive
inverses and they have the same y – coordinate.
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3
Two points are symmetric with respect to the x – axis if
and only if their y –coordinates are additive inverses
and they have the same x – coordinate.
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4
Two points are symmetric with respect to the origin if
and only if both their x – and y – coordinates are
additive inverses of each other.
Imagine sticking a pin in
the given figure at the
origin and then rotating
the figure at 1800. Points
P and P1 would be
interchanged. The entire
figure would look exactly
as it did before rotating.
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5
A function is an even function when f(-x) = f(x) for all x
in the domain of f. This is a function symmetric with
respect to the y – axis.
A function is an odd function when f(-x) = - f(x) for all x
in the domain of f. This is a function symmetric with
respect to the origin.
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6
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2

3
2
2
5
2
x
1
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7
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
cos x
1
2
0
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y y = cos x
3

2



1

2
2

3
2
2
5
2
x
1
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8
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that  1  y  1 .
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
over an x-interval of 2 .
6. The cycle repeats itself indefinitely in both directions of
the x-axis.
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9
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = 2 sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
y = sin x
y = 4 sin x
4
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10
The period of a function is the x interval needed for the
function to complete one cycle.
2
For b  0, the period of y = a sin bx is
.
b
2
For b  0, the period of y = a cos bx is also
.
b
If b > 1, the graph of they function is shrunk horizontally.
period: 2
y  sin x x
y  sin 2 x
period: 


2
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
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11
Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = cos x

2
y = cos (–x)
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12
Steps in Graphing y = a sin bx and y = a cos bx.
1. Identify the amplitude = a .
2
2. Find the period =
.
b
2  1 
1st 
 
b  4
3. Find the intervals.
2  3 
3rd 
 
b  4
2  2 
2nd 
 
b  4
2  4 
4th 
 
b  4
4. Apply the pattern, then graph.
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13
y = a cos bx
 a  b
 max  0  min  0  max
 a  b
 a  b
 min  0  max  0  min
 a  b
y = a sin bx
 a  b
 0  max  0  min  0
 a  b
 a  b
 0  min  0  max  0
 a  b
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14
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
x
y = 3 cos x
y

(0, 3)
2
1

0
3
0

-3
x-int
min
2
max
3
2
0
2
3
x-int
max
(2 , 3)

1 
( , 0)
2
2
3
( , –3)
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2
( 3 , 0)
2
3
4 x
15
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
b
Calculate the five key points.
x
0
y = –2 sin 3x
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2

(  ,-2)
2
2
3

2
5
6

x
(  , 0) 2
3
( , 0)
3
6
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16
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x

x  k  k   
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2

2
 3
2
3
2
x

2
period: 
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17
Steps in Graphing y = a tan bx.
1. Determine the period  .
b
2. Locate two adjacent vertical asymptotes by solving for x:


bx  
2
and bx 
2
3. Sketch the two vertical asymptotes found in Step 2.
4. Divide the interval into four equal parts.
5. Evaluate the function for the first – quarter point, midpoint,
and third - quarter point, using the x – values in Step 4.
6. 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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18
Example: Find the period and asymptotes and sketch the graph
 y

1
x


x

of y  tan 2 x
4
4
3
1. Period of y = tan x is  .

 Period of y  tan 2 x is .
3
 1
2


, 

8
2
 8 3
x
2. Find consecutive vertical
 1
asymptotes by solving for x:
 3 1 
 , 
 , 


 8 3
 8 3
2x   , 2x 
2
2


Vertical asymptotes: x   , x 
4
4


 3
3. Plot several points in (0, )

x
0
2
8
8
8
1
1
1
1

y  tan 2 x 
0
4. Sketch one branch and repeat.
3
3
3
3
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19
Example: Find the period and asymptotes and sketch the graph
1
x 
x   y
of y  3 tan x
2
1. Period of y = tan x is  .
 Period of y  3 tan 1 x is 2
3
2

2. Find consecutive vertical
asymptotes by solving for x:
1
 1

x , x
2
2 2
2
Vertical asymptotes: x    , x  
3. Divide - to  into four equal
parts.
4. Sketch one branch and repeat.
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3
2
2

x
y  3 tan
1
x
2

2
3
0
0

2
3
x
3
2
3
20
1
Graph y  2 tan x
4
x = - 2
y
x = 2

1. Period is 1 or 4.
4
x
2. Vertical asymptotes are
1

1

x   and x 
4
2
4
2
x   2 and x  2
3. Divide the interval - 2
to 2 into four equal parts.
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x
y  2 tan
1
x
4

0
2
0

3
2
2
21
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