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 five x minus pi over two. 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 b x 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 five, and that our h value is associated with pi over two. 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 two. So we're going to have pi over two, and then this is going to be divided by our b value, which I can see is five. So going to pi over two over five, which could also be written as pi over two times five. Because this is just like one over five times pi over two, which would just be pi over two times five. And pi over two times five is pi over 10. So what 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

12. Trigonometric Functions

Graphs of Sine & Cosine

Business Calculus

12. Trigonometric Functions

Graphs of Sine & Cosine

Struggling with Business Calculus?

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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

Step 1: Recall the general form of a cosine function with a phase shift: y = cos(bx - c). The phase shift is determined by the term (bx - c). Specifically, the phase shift is given by c/b, where c is the horizontal shift and b is the coefficient of x.

Step 2: Identify the values of b and c in the given function y = cos(5x - π/2). Here, b = 5 and c = π/2.

Step 3: Calculate the phase shift using the formula phase shift = c/b. Substituting the values, we have phase shift = (π/2) / 5.

Step 4: Simplify the expression for the phase shift. Dividing π/2 by 5 gives π/10. This represents the magnitude of the phase shift.

Step 5: Determine the direction of the phase shift. Since the term inside the cosine function is (5x - π/2), the phase shift is to the right by π/10. If the term were (5x + π/2), the shift would be to the left.

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