# Phase Shift: Definition & Formula

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In mathematics, shifting a function horizontally is referred to as a phase shift. It is one of many ways that we can transform a function. In this lesson, we define phase shift and give a formula for determining its value.

## Phase Shift: A Runner's Analogy

The runners are lined up at the start for the race. Starting precisely when the go signal is given is crucial. Wait, and you are already lagging behind the pack. Start too early, and you are ahead of the pack, but you might get disqualified. These ideas are related to the mathematical concept of phase shift. In this lesson, we will define phase shift and show how it may be determined using a formula.

## Exploring the Basic Idea of Phase Shifting

Returning to the running analogy, if our star runner waits one second before starting to run, she is lagging behind. Let's say that you are standing off to the side and watching the race. From your vantage point the runners are running from left to right on the track. With a one second wait, the star runner is to the left of the pack. What if she waited minus one second? That means she would have started ahead of the gun. She would now be to the right of the pack.

In this instance, the star runner is experiencing a phase shift with respect to the pack. A phase shift is sometimes thought of as a horizontal shift. For a function, a horizontal shift will move the plot over to the left or to the right. This type of change is sometimes called a transformation. Let's plot the function y = x^2, where x is the independent variable:

We see that this curve touches the x-axis at x = 0. This is a good place to focus our attention as we shift the curve. Let's say that we now plot y = (x - 1)^2:

That place where the curve touches the x-axis has shifted to the right. What if we wanted a shift to the left?

That's correct! Letting our function be y = (x + 1)^2 will shift y = x^2 to the left. But just like the runner who's speed is also a factor in determining the phase shift, the phase shift of a function may be more involved.

## Getting More Precise with Phase Shifting

What if we are plotting y = (2x - 1)? Intuitively, you might think that the shift is to the right. That is correct, but the phase shift is not one unit.

The curve has gotten more narrow because each of the points was transformed. We can, however, still see our reference point. The shift is +½ and not the +1 we might be expecting! Here are two ways to predict this:

First, we can ask for the value of x, where we get y = 0. This y = 0 was our reference point in the original function. Then, we algebraically solve for x. This means:

This is like finding the new value of x in our transformed function that produces the same effect as in our original function. In the original function x = 0 now corresponds to x = ½ in the transformed function.

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