Graphing Sine Functions

"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: 

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.

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
____________________________________________________________________________

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)

 Because the period change to , 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!

Graph

1)       Always check for all of the changes first
                                 i.            a => -2, the amplitude is 2, but the graph is flip over x-axis
                                ii.            b => (pi/2) , so period is (2pi/pi/2) = 4, interval is 4/4 = 1
                              iii.            h => (-1/2),  the graph is shifted left by ½
                              iv.            k  => 1, the graph is shifted up by 1

2)        Shift your starting point from the origin to the new location by shifting it 1 unit up and ½ unit left. (-.5   , 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