How to sketch this sin graph?

enter image source here

1 Answer
Feb 26, 2018

See below.

Explanation:

We can express this function in the form:

#y=asin(bx+c)+d#

Where:

  • #color(white)(888)bba# is the amplitude.
  • #color(white)(88)bb((2pi)/b)# is the period. ( this is the normal period divided by #bb(b))#

  • #color(white)(88)bb((-c)/b)# is the phase shift.

  • #color(white)(888)bbd# is the vertical shift.

#y=2sin2(x+pi/6)+1#

Multiply the bracket by the #2#:

#y=2sin(2x+pi/3)+1#

We can see from this, that:

Amplitude is #bb2#

Period is #(2pi)/2=bbpi#

Phase shift is #(-pi/3)/2=-pi/6#

Vertical shift is #1#

If we look at this in relation to #y=sin(x)#, we see that.

The amplitude has been increased by a factor of #bb2#. this causes the graph of #bb(y=sin(x))# to be expanded by a factor of #bb2# in both the positive and negative #bby# direction.

I'll plot all the different changes in relation to #bb(y=sin(x))#

enter image source here

This is shown in the graph #bb(y=2sin(x))#

enter image source here

The period has been compressed by a factor of #bbpi# in the horizontal direction.

enter image source here

The phase shift is #bb(-pi/6)#, this causes the graph of #bb(y=sin(x))# to be translated #bb-pi/6# units in the negative #bbx# direction.

enter image source here

The vertical shift is #bb1#, this causes the graph of #bb(y=sin(x))# to be translated #bb1# unit in the positive #bby# direction.

enter image source here

This is now complete.

Here is a link that explains all the different parts of trig function graphs. It would be worth taking a look. It includes lots of graphics so you can see exactly what's happening.

https://www.mathsisfun.com/algebra/amplitude-period-frequency-phase-shift.html.