How do you find the period and the amplitude for #y = (1/2)sin(x - pi)#?

2 Answers
Jun 3, 2016

#2pi,1/2#

Explanation:

The standard form of the #color(blue)"sine wave function"# is

#color(red)(|bar(ul(color(white)(a/a)color(black)(y=asin(bx+c)+d)color(white)(a/a)|)))#

where amplitude = |a| , period#=(2pi)/b#

c is the horizontal shift and d , the vertical shift.

here a #=1/2,b=1,c=-pi" and "d=0#

hence period#=(2pi)/1=2pi" and amplitude"=|1/2|=1/2#

Jun 3, 2016

#color(blue)("Assumption: period is equivalent to 'pitch'")#

Amplitude: #" "-1/2" to "+1/2#

The period (pitch) is #2pi#

Explanation:

#color(blue)("Amplitude")#

The given equation starts with the basis of sine. This has a maximum and minimum values of -1 to +1.

Making this into a product of #1/2xx"sine"# changes the amplitude to values #-1/2" to "+1/2#
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Period")#

I am interpreting this to be the equivalent of pitch. Like that of a thread. That is the distance between maximum values or any other such repeat.

This standard 'untampered with' period is #2pi# radians or #360^0# if you wish to think in degrees. The way to change this is to apply multiplication on the #x# value in #sin(x)#, say for example #sin(3x)" or "sin(1/2x)#.

As we do not have this structure the pitch remains at the standard #2pi# radians.

Suppose we had #sin(2x)# and suppose we were considering #x=3#.

Imagine that we have two graphs. The one we are drawing and a reference one of #sin(x)#

As we are considering #x=3# we turn to the reference graph. We look at #y =sin(2x) " which is "y=sin(6)# and record that #y# value.

We then turn to the graph we are drawing and plot this recorded #y# value against #x=3#. So in effect we have moved that point to the left from #x=6# to #x=3#. In other words we have reduced the pitch (frequency increase)

So multiplying #x# by #n>1# squashes the graph horizontally and likewise multiplying #x# by #n<1# stretches the graph horizontally.

#color(brown)("We do not have a change in period hear!")#

#color(green)("The "sin(x-pi)" moves the whole plot to the right by "pi" radians")#