# How to find the phase shift and vertical translations of sinusoids?How do I find the phase shift and vertical translation of sinusoid functions? Some problems that I'm having trouble with are...

How to find the phase shift and vertical translations of sinusoids?How do I find the phase shift and vertical translation of sinusoid functions? Some problems that I'm having trouble with are these: 1. Y=-3.5sin(2x-pie/2) 2. Y= 5cos(3x-pie/6) 3. Y=3cos(x+3)-2 I already found out the amptitude and period of these but don't know how to find the phase shift and vertical translations of these sinusoids. Thanks in advance!=)
nathanshields | Certified Educator

Hi -

Think back to linear equations in the form y = mx + b.  The y-intercept (b) is the up/down (vertical) translation of the line y = mx.

So, consider y = 3cos(x+3).  It's going to look like all squiggly, right?  Now throw in a vertical translation: y = 3cos(x+3)-2.  All the y-values have now just dropped by 2, so the whole graph shifts down 2 units.

Keep your eye out for a constant (number) being added or subtracted from the sinusoid.  For example, y = sin(3x)+4 has a vertical translation of positive 4 ("shift up 4").

Now on to phase shift.  Just as adding a constant to the y value shifts your graph up or down, adding a constant to the x variable shifts the graph left and right (this is called phase shift).  The only tricky part is that you may need to factor first:

sin(2x-pi/2) has a constant being subtracted from 2x.  To see what's being subtracted from x, we must factor:
sin(2x-pi/2) = sin(2(x-pi/4))
So the phase shift here is pi/4 units to the... right.  If a value is being added to x, the graph moves left, and vice versa (to understand this, make some quick linear graphs: y = 2(x+c) for different values of c).

EX: y = cos(4x - pi) - 1.
First the easy part: vertical translation of 1 unit down.
Now factor the argument: y = cos(4(x - pi/4)) - 1.  horizontal translation ("phase shift") of pi/4 units right.

I hope this helps!

sciencesolve | Certified Educator

The phase shift is called horizontal translation, hence the sinusoid is shifted to the left or to the right when the constant c is added to or subtracted from the argument of the trigonometric function.

If c>0, then the sinusoid is shifted to the right and if c<0, the sinusoid is shifted to the left.

If y=-3.5sin(2(x-`pi/2` )) => the phase shift is c = `pi/2`  and the sinusoid is shifted to the right.

If y=5cos(3(x-`pi/18` ))=> the phase shift is c = `pi/18 ` and the sinusoid is shifted to the right.

If y=3cos(x+3) - 2=>the phase shift is c = -3 and the sinusoid  is shifted to the left and the vertical translation is down d = -2 units.

Hence, the phase shifts are: 1)c=`pi/2 ` ; 2) c = `pi/18`  ; 3) c = -3 and the vertical translations are )d=0; 2) d = 0 ; 3) d= -2.