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Properties of the Sine Curve
1. The period is 360.2. The amplitude is 1.
3. The curve is continuous.4. y=0 when x= 180 and 360.
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x
y
-8
-6
-4
-2
0
2
4
6
y=sin x
y=2sin x
y=5sin x
-a sine graph has ashape of an inverse
letter s
-From diagram 1.1,
the graph of y= sin x
has an amplitude of1 ;the graph of
y= 5sin x shows an
amplitude of 5.
-the coefficient of
sine indicates the
amplitude of the
graph.
Diagram 1.1
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x
y
-3
-2
-1
0
1
2
3
y=2sin x
y=2sin 2x
Coefficient of
x > 0:
-the
coefficient of xis positive, the
graph is in
original state.
-e.g, the
graph of
y=2sin2xportrayed two
periods instead
of one.
-the
coefficient of xindicates the
number of
period in a
graph.
Diagram 1.2
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x
y
-2
-1
0
1
2
3
4
5
y=2sin x
y=2sin x+1
y=2sin x+2
Effect of c:-the graph will
shift up when a
number is
added to the
equation.-noticed that
the number
added is the y-
intercept of the
graph.
-the number
added = c = y-
intercept.
Diagram 1.3
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x
y
-4
-3
-2
-1
0
1
2
3
y=2sin x
y=2sin x-1
y=2sin x-2
Effect of c:
-the graphs shift
to the bottom
when a negative
number is added
to the equation.
e.g, the graph
of y=2sin x-2 cut
y-axis at (0, -2).
-same effect of
the graph as theprevious page.
Diagram 1.4
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x
y
-8
-6
-4
-2
0
2
4
6
y=(2sin x)/1
y=(2sin x)/6
y=(2sin x)/2
Decrease in
amplitude:
-divided whole
equation by a number
which is >0, the graphwill has lower and
lower amplitude until
it becomes a straight
line.
-e.g, y= (2sin x)/2 is a
graph with amplitude
of 1.
-the number is
dividing the
amplitude of the
graph.- the amplitude of a
graph is influenced by
the number which
divided the whole
equation.Diagram 1.5
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x
y
-8
-6
-4
-2
0
2
4
6
y=(2sin x)
y=(2sin x)*2
y=(2sin x)*4
Increase in
amplitude:-when the whole
equation is being
multiplied by a
number which is >0,
the amplitude of thecurve will increase
double or triple as
well.
- the amplitude of a
graph increase whenbeing multiplied.
Diagram 1.6
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x
y
-8
-6
-4
-2
0
2
4
6
y=2sin x y=2sin -x
Coefficient of x < 0:
-coefficient of x is
negative, the graph
inverted or
reflected to theopposite side.
-a graph will be
inverted when the
equation is
negative.
Diagram 1.7
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x
y
-8
-6
-4
-2
0
2
4
6
y=2sin (x/4)
y=2sin (x/1)
y=2sin (x/2)
Decrease in period:
-When x is dividedby a number which
is >0, for instance,
the graph of y=2sin
(x/2) shows only
half period in thegraph.
-The first period
will end at 4.
-divide the x value
to reduce thenumber of period.
Diagram 1.8
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x
y
-8
-6
-4
-2
0
2
4
6
y=2sin (x/-4) y=2sin (x/-1)
y=2sin (x/-2)
Decrease in period
(reflected):
-When x is divided
by a number which
is
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x
y
-8
-6
-4
-2
0
2
4
6
y=|-4sin x|
y=|-2sin x| y=|-sin x|
x
y
-8
-6
-4
-2
0
2
4
6
y=|-4sin -x|
y=|-2sin -x| y=|-sin -x|
The effects of modulus
-modulus results in a
positive answer.-From the graph, all the
curve had been
reflected up to the x-
axis.
- E.g, the graph of y=|-sin x| shows a period
with amplitude 1
instead of -1.
-the 2nd graph also
shows the same pattern
although x is negative.
-modulus makes the
negative points to
become positive.
Diagram 1.11
Diagram 1.10
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Properties of the Cosecant Curve
1. The period is 360o
or 2
.2. The graph is symmetrical about the origin.
3. There are vertical asymptotes at x = -2 , - , 0, , 2 , etc.
4. There are turning points at ( /2, 1), (- /2, -1), etc
5. The curve is not defined for the range -1 < y < 1.
6. The curve is not continuous.
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-divided the whole equation by a positive number, the turning
point will becomes wider. The larger the positive number as
divisor, the wider the turning point of the curve. The divided
number only affects the shape of the turning point.
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x
y
-8
-6
-4
-2
0
2
4
6
y= (1/sin x)/-5
y= (1/sin x)/-3
y= (1/sin x)/-1
-divided the whole equation by a negative number, each curve will
move to the opposite side like wise the left one move to the right side
and vice versa. The curve of the turning point becomes wider and
wider when divided by a bigger negative value.
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-x is deducted by a positive number, the curve will move to the
right The range is depending on the value of the positive number.
The larger the number, the bigger the range of the curve move to
the right. When x is added by a positive number, this situation isopposite which means the curve will move to the left.
x
y
-6
-4
-2
0
2
4
6
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