phase shift

The form given by the text for a sine function is

• y = A sin( B t - C ) + D.

Actually the text uses x instead of t, which is fine. The variable can be anything you want to call it--x, t, z, phi, theta, or aardvark if you like.

However I prefer to use t when first introducing this form, and to relate the function to motion in time around a circle. Using x we run the risk of confusing the independent variable x of the given form with the x coordinate of the point on the unit circle, which is something very different.

Now from Section 1.5 you should know that if y = f(t) is a function, which we assume to be represented by both a table and a graph, then we can characterize horizontal and vertical shifts, vertical stretch and horizontal compression as follows.

• A table for y = A * f(t) vs. t can be made from the table for y = f(t) vs. t by multiplying all y values by A.
• The effect of this is to vertically 'stretch' the graph of y = A * f(t) so that every point of y = A * f(t) is | A | times as far away from the t axis as the corresponding point of y = f(t). If A is negative the points will also lie on opposite sides of the t axis.
• A table for y = f(t) + c can be made from the table for y = f(t) vs. t by adding c to all y values by A.
• The effect of this is to vertically 'shift' the graph of y = f(t) + c so that every point of y = f(t) + c is displaced c units in the vertical direction from the corresponding point of y = f(t). If c is negative the new graph will lie lower than the original; if c is positive the new graph will lie higher. The vertical distance between a point on the graph of y = f(t) and the graph of y = f(t) + c is always the same, which is not the case with a vertical shift.
• A table for y = f(t - h) can be made from the table for y = f(t) by subtracting h from every value of t. The result is that to get the same value of y = f(t - h), the value of t must be h units greater than for the function y = f(t).
• The graph of y = f(t - h) will therefore be shifted h units (not - h units, but h units) in the horizontal direction from the graph of y = f(t).  We say the graph has horizontal shift h.
• A table y = f(k t) can be made from the table for y = f(t) if we multiply every value of t by k. The result is that to get the same value of y = f( k t ), the value of t must be 1/k times as great as for the function y = f(t). That is, y = f(kt) gives us the same y value for 1/k times the t value.
• The graph of y = f(k t) will therefore by 'compressed' relative to the graph of y = f(t), by factor k in the horizontal direction. Every point of the transformed graph will lie k times closer to the t axis.  We say that the graph has been horizontally compressed by factor k.

y = A sin( B t - C) + D can therefore be constructed from a graph of y = A sin(t) by first taking the following steps:

• Vertically stretch the graph by factor A to get the graph of y = A sin(t).
• Vertically shift the graph of y = A sin(t) by D units to get the graph of y = A sin(t) + D
• Horizontally compress the graph of y = A sin(t) + D by factor B to get the graph of y = A sin(B t) + D.

This still leaves us the horizontal shift, and here we have to be careful.  We might be tempted to use h = C for the horizontal shift, but the horizontal shift must be done by replacing t with t - h.  If we were to replace t by t - C, the function y = A sin(B t + C) + D will become y = A sin(B ( t - C)) + D, which would be equal to A sin(B t - B C) + D, not to A sin( B t - C) + D.

To see what the horizontal shift is, we have to first factor out B in the expression B t - C.  We get B t - C = B ( t - C / B).  Writing our function using this factored form we have

• y = A sin(B ( t - C / B) ) + D.

Now if we replace t in y = A sin(B t ) + D with t - C / B, we will horizontally shift the graph C / B units.  So our last step in constructing the graph of y = A sin(B t - C) + D is this horizontal shift.

The final graph will therefore start from the graph of y = sin(t) and will be

• vertically stretched A units, then
• vertically shifted D units, then
• horizontally compressed by factor B, and finally
• horizontally shifted C / B units.

This will result in a graph with amplitude A, vertically centered about the horizontal line y = D, with period change from the original 2 pi of the sine function to 2 pi / B by the horizontal compression, and horizontally shifted C / B units.

This final horizontal shift is called the phase shift.  Phase shift is nothing but the horizontal shift.

A graph which starts with period 2 pi and is horizontally compressed by factor B ends up with period 2 pi / B.

Relationship to Circular Motion

A graph of the y coordinate of a point moving on a circle of radius A, with the circle centered at position (h, k), with the radial line originally at angle phi, and moving with angular velocity omega is

• y = A sin( omega ( t + phi) ) + k.

A graph of this y coordinate vs. clock time t has amplitude A, horizontal compression factor omega, vertical shift k and phase shift -phi.

The period of the motion is 2 pi / omega.