Describe the phase shift for the following function:y=cos(5x−π2)y=\cos\left(5x-\frac{\pi}{2}\right)y=cos(5x−2π)

π 10 \frac{\pi}{10} 10 π to the right

Describe the phase shift for the following function: y = cos ( 5 x − π 2 ) y=\cos\left(5x-\frac{\pi}{2}\right) y = cos ( 5 x − 2 π )

Hey, everyone. Let's see how we can solve this problem. So here we're asked to describe the phase shift for the following function, y is equal to the cosine of 5x minus pi over 2. Okay. Now, to solve this problem, what we need to do is recognize that the general form that we're going to have in these types of situations where we have this cosine is going to be a times the cosine of bx minus h plus k. This is the basic form that we see for this. And the reason that we need to know this is because we need to make sure that everything fits for the phase shift depending on what we have in here. Now I can see that we have this minus sign, and then our b value is associated with 5, and that our h value is associated with pi over 2. So this all looks good. Now, recall that for the phase shift, the way that we can determine this is by using the ratio h over b. This tells us how much our graph is going to have shifted either to the left or right. Now what our h value is that I can see here is pi over 2. So we're going to have pi over 2, and then this is going to be divided by our b value, which I can see is 5. So going to pi over 2 over 5, which could also be written as pi over 2 times 5, because this is just like 1 over 5 times pi over 2, which would just be pi over 2 times 5. And pi over 2 times 5 is pi over 10. So we can do is say that our graph shifted by pi over 10 units, but the question is, is this to the left or to the right? Well, because we got a positive h value, that means our graph is going to shift to the right. So this is going to be pi over 10 units to the right, and this is how we could describe the phase shift of this cosine function. So this right here would be the solution to our problem. So that's how you can solve it.

Describe the phase shift for the following function: y = cos ⁡ ( 5 x − π 2 ) y=\cos\left(5x-\frac{\pi}{2}\right) y = cos ( 5 x − 2 π )

π 10 \frac{\pi}{10} 10 π to the right

π 2 \frac{\pi}{2} 2 π to the right

π 2 \frac{\pi}{2} 2 π to the left

π 10 \frac{\pi}{10} 10 π to the left

4. Graphing Trigonometric Functions

Phase Shifts

Trigonometry

4. Graphing Trigonometric Functions

Phase Shifts

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Multiple Choice

Describe the phase shift for the following function:
$y=\cos\left(5x-\frac{\pi}{2}\right)$

$\frac{\pi}{2}$ to the right

$\frac{\pi}{2}$ to the left

$\frac{\pi}{10}$ to the right

$\frac{\pi}{10}$ to the left

Verified step by step guidance

Identify the general form of the cosine function with a phase shift: y = cos(bx - c), where the phase shift is given by c/b.

In the given function y = cos(5x - \frac{\pi}{2}), compare it to the general form to identify b and c. Here, b = 5 and c = \frac{\pi}{2}.

Calculate the phase shift using the formula \text{Phase Shift} = \frac{c}{b}. Substitute the values of c and b: \text{Phase Shift} = \frac{\frac{\pi}{2}}{5}.

Simplify the expression for the phase shift: \text{Phase Shift} = \frac{\pi}{10}.

Determine the direction of the phase shift. Since the expression inside the cosine function is (5x - \frac{\pi}{2}), the phase shift is to the right by \frac{\pi}{10}.

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