Describe the phase shift for the following function:y=cos(2x+π6)y=\cos\left(2x+\frac{\pi}{6}\right)y=cos(2x+6π)

π 12 \frac{\pi}{12} 12 π to the left

Describe the phase shift for the following function: y = cos ( 2 x + π 6 ) y=\cos\left(2x+\frac{\pi}{6}\right) y = cos ( 2 x + 6 π )

Hey everyone. So let's give this problem a try. Here we're asked to describe the phase shift for the following function y is equal to the cosine of 2x plus pi over six. So how can we go about solving this? Well what we need to do is recognize the most general form for a cosine, which would be a times the cosine of bx minus h plus k. Now looking at our function, I can see that it doesn't appear we have any kind of number out front here. So we can ignore any kind of changes in the amplitude. And I also see that we don't have anything added to the outside here. So we can ignore this plus k. So this is really the inside of our function that we're looking at. And we need to make sure that this form matches. Now one thing I'm noticing is that we have a difference in the sign. I notice we have a minus sign right there, whereas we have a plus sign up here. So to get this into the most general form, we need to force a minus sign in some kind of way. But how can we do that? Well, what we can do is recognize that y is equal to the cosine of two x plus pi over six is the same thing as y is equal to the cosine of two x minus negative pi over six. Now the reason that we're writing it like this is because typically we would cancel these negative signs, but this now allows us to have the correct form where we have a minus sign right here. So what I can see from what we have is that our h is going to be this value, which is negative pi over six. So we have negative pi over six for h, and then what I can see is this number out front here, our b value is equal to two. Now this is going to tell me how I can describe the phase shift for this function, because recall the phase shift is just going to be h or over b units to the left or right. Well, our h is negative pi over six, we just figured that out, and our b value is two. So negative pi over six over two could also be written as negative pi over six times two. And six times two is 12, so we're gonna have negative pi over 12. So this right here is our ratio h over b, and what that means is that we can see since we have pi over 12, that means that our graph has gone over a phase shift for pi over 12 units. But is this going to be pi over 12 units to the right or to the left? Well, notice we got a negative sign here. And because we got a negative sign, that means we have gone pi over 12 units to the left. So this right here would be the best description of our phase shift, and that's the solution to this problem.

Describe the phase shift for the following function: y = cos ⁡ ( 2 x + π 6 ) y=\cos\left(2x+\frac{\pi}{6}\right) y = cos ( 2 x + 6 π )

π 12 \frac{\pi}{12} 12 π to the left

π 6 \frac{\pi}{6} 6 π to the right

π 6 \frac{\pi}{6} 6 π to the left

π 12 \frac{\pi}{12} 12 π to the right

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:
$y=\cos\left(2x+\frac{\pi}{6}\right)$

$\frac{\pi}{6}$ to the right

$\frac{\pi}{6}$ to the left

$\frac{\pi}{12}$ to the right

$\frac{\pi}{12}$ 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). To find the phase shift, rewrite the equation in the form y = cos(b(x + c/b)).

Step 2: In the given function y = cos(2x + π/6), identify the coefficient of x (b = 2) and the constant term (c = π/6).

Step 3: Factor out the coefficient of x (b = 2) from the term (2x + π/6). This gives y = cos(2(x + (π/6)/2)). Simplify the fraction (π/6)/2 to get π/12. The equation becomes y = cos(2(x + π/12)).

Step 4: The phase shift is determined by the term (x + π/12). Since the term inside the parentheses is (x + π/12), the phase shift is π/12 to the left. A positive value inside the parentheses indicates a shift to the left.

Step 5: Conclude that the phase shift for the function y = cos(2x + π/6) is π/12 to the left.

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