Horizontal Shift - Phase Shift
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The graphs of sine and cosine are the same when sine is shifted left by .   Such a shifting is referred to as a horizontal shift. 
shift sine to the left to create cosine
shift cosine to the right to create sine




horizontal shift and phase shift

If the horizontal shift is positive, the shifting moves to the right.  If the horizontal shift is negative, the shifting moves to the left.  

From the sinusoidal equation, ,
the horizontal shift is obtained by determining the change being made to the x value.
The horizontal shift is C.
When the value B = 1, the horizontal shift, C, can also be called a phase shift, as seen in the diagram at the right.

The easiest way to determine horizontal shift is to determine by how many units the "starting point" (0,0) of a standard sine curve, y = sin(x), has moved to the right or left.

Horizontal Shift or


Why does move left and move right?
Doesn't "+" mean right???  What's happening here???

Be careful here!!  When trying to determine the direction of a horizontal shift, be sure you have the function in the proper form, .  (Notice the subtraction of C.)
The horizontal shift is determined by the value of C.

 This expression is really where the value of C is negative and the shift is left.
 In this expression the value of C is positive and the shift is right.

Another BEWARE!!!

What is the horizontal shift of ?

 At first glance, it may seem that the horizontal shift is , but it is NOT.

 The horizontal shift is actually , because the equation is actually


Physicists and engineers say that the "phase shift" in the above problem is actually
They make a distinction between y = Asin(B(x - C)) + D and y = Asin(Bx - C) + D,
calling the phase shift C in the first equation when B = 1, otherwise calling it C in the second equation.
While this distinction exists for physicists and engineers, some mathematics textbooks use the terms "horizontal shift" and "phase shift" to mean the same thing.  Refer to your textbook, or your instructor, to see what definition they wish you to use for "phase shift", if they are using that term.  We will be discussing "horizontal shifts".


Horizontal shifts can be applied to all trigonometric functions.



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