How do you sketch #y=sin (x/2)#?

1 Answer

Answer:

Using a 6 point summery.

Explanation:

A six point summery is identifying 6 major facts about the graph of a trig function is as followed.

#1.# Amplitude

This is simply how high a graph will go. This is found out by looking at the coefficient in front of the trig function. In the graph of #y=sin(x/2)#, there appears none but that is because it is actually #1#, and mathematicians don't write the number one out most times(if at all).

Thus, this graph will have an amplitude of #1#, which means its greatest height is 1 and lowest height is #-1#.

#2.# Period

The parent sine graph, #y=sin(x)#, has a period of #2pi# since it takes #2pi# revolutions to complete one cycle of the graph.

To figure out the new period, simply divide #2pi# by the coefficient attached to your #x# value. In this case, the new Period is #4pi#, since the coefficient is one half.

#3.# Translation of the Graph Up or Down

This is simply the question will my graph move up, down, or not at all. Again the parent sine function #y=sin(x)# doesn't have a translation, but if it were #y=sin(x) + 1# it would be moved one unit up. This graph doesn't have any, but this is part of the process

#4.# Translation of the Graph Left or Right

This graph doesn't have any but it is also important to note. Lets say our function we must graph is #y=sin(x+pi)#. The entire graph would be shifted to the left one #pi# units. It shifts to the right because in order for the function to be #0#, it needs to have an #x# value of #-pi#.

#5.# Five Number Summery

A five number summery is simply #5# points on your graph used to map out what you will draw. I detailed it in the following paragraph.

A sine curve #sin(x)# starts at the origin #(0,0)#, has a maximum of #1# at #x= pi/2#, a zero at #x=pi#, a minimum of #-1# at #x= (3pi)/2#, and a zero at #2pi#. What I just did right there is a five number summery, where I identify.

#6.# Consider range and domain

This is huge when dealing with other trig functions. In this case the domain is all real numbers and the range is #-1# to #1#, written mathematically #[-1,1]#.

The graph of #sin(x)# looks like this

graph{sinx [-10, 10, -5, 5]}

The graph of #sin(x/2)# looks like this

graph{sin(x/2) [-10, 10, -5, 5]}