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High School Precalculus: Help and Review32 chapters | 297 lessons

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.

**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 (2*x* - 4) + 6. All you have to do is follow these steps.

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.

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 (2*x* - 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 (2*x* - 4), you would need to rewrite your function so your addition of 6 is at the end: *y* = sin (2*x* - 4) + 6.

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.

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.

For your example function, you have your C value as 4 and your B value as 2. Dividing the C value by the B value, you get 4 / 2 = 2.

So your phase shift is 2. This means that your function is shifted 2 units to the right. And you are done. Your answer is 2.

Let's try another example.

Find the phase shift for the function *y* = 3 cos (2*x* + 8).

Looking at this function, you see that it is already in standard form, so you can go straight to step 2 and label your values. Comparing your function to the standard form, you see that your A is 3, your B is 2, your C is -8, and your D is 0. Your C is -8 because it is being added in your function. In order for your standard form to have a C that is being added, the C needs to be negative.

To calculate your phase shift, you remember the only two numbers you need to be concerned about are your B and C values because the calculation for phase shift is C / B. Making this calculation, you find your phase shift to be -8 / 2 = -4. Your function is shifted 4 units to the left. A negative phase shift means the function is moving to the left, and a positive phase shift means the function to moving to the right.

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High School Precalculus: Help and Review32 chapters | 297 lessons

- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Graphing the Cosecant, Secant & Cotangent Functions 7:10
- Using Graphs to Determine Trigonometric Identity 5:02
- Solving a Trigonometric Equation Graphically 5:45
- How to Graph cos(x) 5:10
- How to Graph 1-cos(x) 6:15
- How to Find the Period of a Trig Function 4:19
- How to Find the Phase Shift of a Trig Function
- How to Find the Frequency of a Trig Function 4:59
- Go to Trigonometric Graphs: Help and Review

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