"Graphing sine is as easy as transforming the parent sine graph y=sinx around."
____________________________________________________________________________
As you may have remembered from geometry and trig, the sine of an angle is it's opposite side over the hypotenuse side. And in addition you have also learned about the reference
angles and the unit circle, such as sin(300)
= ½, or
sin (pi/2) = 1, etc… Want a quick recap?
Here's an example:
sin (pi/2) = 1, etc… Want a quick recap?
Here's an example:
Let's take this equation y = sinx and make a table with common points
x
|
0
|
pi/2
|
pi
 |
3pi/2
|
2pi
|
Sin(x)
|
0
|
1
|
0
|
-1
|
0
|
Draw the points, connect the dots, and graph...
Here, we see the basic sine
graph: y=sin x known as the
“parent graph of sinx.
Here is a more general form:
The amplitude,
the distance from x-axis and the “hump”, is 1.
The period, the part unique part of the graph that was repeated, is.
The period, the part unique part of the graph that was repeated, is
.
o Notice how at each fourth section of
the period – interval, we plot our point. So each interval is 2pi/4, which is pi/4.
o Sine graph characteristic: 1)go right by interval, move up by amplitude
2)go right by interval, stay
3)go right by interval, move down by amp.
4)go right by interval, stay
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____________________________________________________________________________
In y=asin(bx - h) + k, a represents the vertical
stretch of
the graph, which is also used to tell the amplitude or height of the “hump”.
·
Graphs of
y=sin(x) vs. y =2sin(x) vs. y= -sinx
x
|
0
|
pi/2
|
pi
|
3pi/2
|
2pi
|
2Sin(x)
|
0
|
2
|
0
|
-2
|
0
|
-Sin(x)
|
0
|
-1
|
0
|
1
|
0
|
Notice when a = 2, the vertical stretch is 2 and thus, the amplitude became 2. However, when a = -1, the vertical stretch become -1, causing the graph to “flip” over the x-axis.
b
represents the horizontal stretch of
the graph, a change in b will affect the period. To FIND the new period, take 2pi and divide by b to get
the new period.
·
Graphs of y=sin(x) vs. y=sin(2x)
x
|
0
|
pi/4
|
pi/2
|
3pi/4
|
pi
 |
Sin(2x)
|
0
|
1
|
0
|
-1
|
0
|
Notice when b =2(increased), the period is (decreased)
(decreased)
Because the period change to 
, The new
interval is
, The new
interval is
In the parent graph, h
represents the horizontal shift of
the graph; it helps tell how many units the graph shift, and whether it’s right
of left.
·
Graphs of
y=sin(x) vs. y =sin(x -)
)
x
|
pi
|
3pi/2
|
2pi
|
5pi/2
|
3pi
|
Sin(x)
|
0
|
1
|
0
|
-1
|
0
|
Since h became pi,
the graph will shift pi unit to the right
k
represents the vertical shift of the
graph; it helps tell how many units the graph shift, and whether it move up or
down.
·
Graphs of y=sin(x) vs. y=sin(x) -1
x
|
0
|
pi/2
|
pi
|
3pi/2
|
2pi
|
Sin(x)
|
-1
|
0
|
-1
|
-2
|
-1
|
Since k became (-1), the graph will
shift 1 unit down.
Let's Graph Some EXAMPLES!
i. a => -2, the amplitude is 2, but the graph is flip over x-axis
ii. b => (pi/2)
iii. h => (-1/2), the graph is shifted left by ½
iv. k => 1, the graph is shifted up by 1
3) Next, use the sine characteristic to plot the rest of the points. Interval and amplitude are very very helpful in this stage.
i. Move right by 1, go down 2
ii. Move right by 1, stay at 1
iii. Move right by 1, go up 2
iv. Move right by 1, stay at 1
4)
Draw the
graph by connecting the points & check if the amplitude and the period
match that of the original problem.-TrigInADay
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