How do you graph # y=1/2+2sin(3x-pi/2)#?

1 Answer
Oct 4, 2016

Answer:

The amplitude is #2#, the period is #(2pi)/3#, the phase shift is #pi/6# right, and the vertical shift is #1/2#.

Explanation:

First, rewrite the problem as #y=2sin(3x-pi/2) +1/2#

This is of the form #y=Asin(Bx-C)+D# where

#A=# the amplitude

#(2pi)/B=# the period

#C/B=# the phase shift

#D=# the vertical shift

In this example, the amplitude #A=2#. Amplitude is the vertical distance from the "midline" to the max or min. It is not the the distance from max to min.

Period #=(2pi)/B=(2pi)/3#. One complete cycle of the sin graph will be #(2pi)/3# horizontal units wide.

Phase shift #=C/B=frac{pi/2}{3}=pi/6# The graph will be shifted #pi/6# units to the right.

Vertical shift #D=1/2# units up.

Let's look at each transformation of the graph. First #y=sinx#

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Next, change the amplitude to #2#
enter image source here

Now change the period from #2pi# to #(2pi)/3#

enter image source here

Next, add a phase shift of #pi/6# to the right. The red graph has the phase shift. The blue is without the phase shift.
enter image source here

Lastly, add a vertical shift of #1/2# units up.
enter image source here