# 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...

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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!

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.**