Download Graphs of Trigonometric Functions

Digital Lesson
Graphs of Trigonometric
Functions
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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2
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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3
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 = cos x
y
3

2



1

2
2

3
2
2
5
2
x
1
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4
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
0

-3
x-int
min
2
max
(0, 3)
2
1
1
2
3


(  , 0)
2
2
( 3 , 0)
2
3
2
2
3
0
x-int
(2 , 3)
max
3
4 x
( , –3)
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5
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 = 2sin 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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6
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 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If b > 1, the graph of the
y function is shrunken horizontally.
y  sin 2 x
period: 2
y  sin x 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 
3

4

2

x
period: 4
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7
Example : 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
x
y = sin x
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
2
8
Example: Sketch the graph of y = - 2 sin (3x).
The function in the form y = a sin bx with b > 0
y = - 2 sin (–3x)
2  2
period:
amplitude: |a| = |–2| = 2
=
b
3
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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9
y  a sin(bx  c)  d
a = the height between the max/min and
the midline. If negative, the graph
reflects over its midline
b = affects the period. In fact,
2
period 
b
c = affects the phase shift (i.e. horizontal).
Shifts graph in opposite direction of the
c
phase

shift

sign in the parentheses.
b
d = affects the vertical shift. It moves the
midline up (if positive) and down (if
negative).
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10
Sketching A Sinusoid
• Check the midline (d) and sketch it.
• Find the amplitude and note how high/low
the graph will go
2π
𝑏
• Find the period = and break up the
midline into quarters.
• Start your graph at 𝑏𝑥 − 𝑐 = 0
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11
Examples
• 𝑦 = 2 sin 2𝑥 − π − 2
• 𝑦=
1
𝑥
− cos
2
2
+
π
2
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+3
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