SINE Simplified)

8/2/2019 SINE Simplified)

1/19


2/19

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.


3/19

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


4/19

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


5/19

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


6/19

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


7/19

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


8/19

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


9/19

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


10/19

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


11/19

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


13/19

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.


14/19


15/19


16/19

-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.


17/19

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.


18/19

-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


19/19

SINE Simplified)

Documents

Transcript of SINE Simplified)