How to Find the Phase Shift of a Trig Function

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

As long as your trig function is written in standard form, you can easily find your phase shift. You just need to know which two numbers to look at and how to combine them.

The Steps

Trig functions are functions of angles. Usually, you'll see your trig functions include either a sine, cosine, tangent, or cotangent. When it comes to evaluating trig functions, finding the phase shift is one type of problem that you need to know how to solve. The phase shift is how far the function is shifted horizontally either to the right or left. It might sound difficult to find, but it's actually quite easy.

Say you needed to find the phase shift for the trig function y = sin (2x - 4) + 6. All you have to do is follow these steps.

Step 1: Rewrite your function in standard form if needed.

The first you need to do is to rewrite your function in standard form for trig functions. You'll see later on how this makes your life so much easier!

The standard form for trig functions is this one.

trig phase shift

The A stands for the function's amplitude. The B is used to calculate the period. The D gives you the vertical shift. Your phase shift is C / B. You can replace the sine with any of the other trig operations such as cosine, tangent, and cotangent.

If you look at the function you need to find the phase shift for, y = sin (2x - 4) + 6, it looks like it's already in standard form so you don't need to rewrite it.

If your function is not in standard form, you'll need to rewrite your function so it is. For example, if you had y = 6 + sin (2x - 4), you would need to rewrite your function so your addition of 6 is at the end: y = sin (2x - 4) + 6.

Step 2: Label your values.

The second step, after your function is in standard form, is to label your A, B, C, and D values. Be careful here when labeling your C value. Because the standard form is subtracting the C, if your C is also being subtracted, then your C value will be positive, but if your C is being added, then your C value will be negative.

Comparing your function to the standard function, you can see that your A = 1, your B = 2, your C = 4, and your D = 6.

Step 3: Calculate the phase shift.

Your third and final step is to calculate your phase shift. Remember that the phase shift, from your function in standard form is C / B. All you have to do is to plug in your values for C and B. The other values, A and D, don't matter. If you remember this, then the only two numbers you need to look at are your C and B value.

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