Graphs of trigonometry functions

1. Graphs of Trigonometric
Functions
Digital Lesson

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6. The cycle repeats itself indefinitely in both directions.
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
1. The domain is the set of real numbers.
5. Each function cycles through all the values of the range
over an x-interval of .p2
2. The range is the set of y values such that . 11££-y

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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.
0-1010sin x
0x
2
p
2
3p
p2p
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
2
3p
-
p- 2
p
-
p22
3p
p2
p
2
5p
1-
1
x
y = sin x

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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.
10-101cos x
0x
2
p
2
3p
p2p
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
2
3p
-
p- 2
p
-
p22
3p
p2
p
2
5p
1-
1
x
y = cos x

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y
1
1-
2-
3-
2
x
p p3p2p- p4
Example: Sketch the graph of y = 3 cos x on the interval [–p, 4p].
Find the key points; graph one cycle; then extend the graph in both
directions for the required interval.
maxx-intminx-intmax
30-303y = 3 cos x
2pp0x 2
p
2
3p
(0, 3)
2
3p
( , 0)
( , 0)
2
p
p2( , 3)
p( , –3)

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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 there is a negative in front (a < 0), the graph is reflected in the x-
axis.
2
3p
2
p
4
y
x
4-
p2p
y = – 4 sin x
reflection of y = 4 sin x y = 4 sin x
y = sin x
2
1
y = sin x
y = 2 sin x
When I ask for
amplitude I will not
ask what kind of
stretch it is. Instead,
I will ask for the
value of the
amplitude.

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y
x
p- p2p
sinxy=
pperiod: 2 2sinp=y
pperiod:
The period of a function is the x interval needed for the
function to complete one cycle.
For b > 0, the period of y = a sin bx is .
b
p2
For b > 0, the period of y = a cos bx is also .
b
p2
If 0 < b < 1, the graph of the function is stretched horizontally.
If b > 1, the graph of the function is shrunk horizontally.
y
x
p- p2p p3 p4
cosxy=
pperiod: 2
2
1
cosxy=
pperiod: 4

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y
x
p2p
y = cos (–x)
Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
Use the identity
sin (–x) = – sin x
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
Example 2: Sketch the graph of y = cos (–x).
Use the identity
cos (–x) = – cos x
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
x
p2p
y = sin x
y = sin (–x)
y = cos (–x)

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2
y
2-
6
p
x
2
p
p6
5p
3
p
3
2p
6
p
6
p
3
p
2
p
3
2p
020–20y = –2 sin 3x
0
x
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
amplitude: |a| = |–2| = 2
Calculate the five key points.
(0, 0)
( , 0)
3
p
( , 2)
2
p
( , -2)
6
p
( , 0)
3
2p
Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x
period:
b
p 2p 2
3
=

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y
x
2
3p-
2
3p
2
p
2
p-
Tangent Function
Graph of the Tangent Function
2. range: (–¥, +¥)
3. period: p
4. vertical asymptotes:
( )p
pp
ryrepeatsevex
2
3
,
2
=
1. domain : all real x
( )ZÎ+¹ kkx
2
p
p
Properties of y = tan x
period: p
To graph y = tan x, use the identity .
x
x
x
cos
sin
tan=
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.

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2. Find consecutive vertical
asymptotes by solving for x:
4. Sketch one branch and repeat.
Example: Find the period and asymptotes and sketch the graph
of xy 2tan
3
1
=
2
2 ,
2
2
pp
=-= xx
4
,
4
pp
=-= xxVertical asymptotes:
)
2
,0(
p
3. Plot several points in
p
1. Period of y = tan x is .
2
p
. is 2tan of Period xy=®
x
xy 2tan
3
1
=
8
p
-
3
1
- 0
0
8
p
3
1
8
3p
3
1
-
y
x
2
p
8
3p
-
4
p
=x
4
p
-=x
÷
ø
ö
ç
è
æ
-
3
1
,
8
p
÷
ø
ö
ç
è
æ
3
1
,
8
p
÷
ø
ö
ç
è
æ
-
3
1
,
8
3p