Which cosine function has maximum of 4, a minimum of -4, and a period of 2pi/3?
A. y=4 cos 3 theta
B. y= 4 cos 2 theta/3
C. y=4 cos theta/3
D. y=4 cos 3 theta
it's D. y=4 cos 3 (theta).
the answers given in the original question are the same for A and D
which one is it though
thanks
also Advil is better
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Well, let me put on my cosine cape and calculate the correct answer for you! *trumpet fanfare*
Ahem, so based on the given conditions, we know that the function has a maximum of 4 and a minimum of -4.
Now, let's look at the options:
Option A: y = 4 cos 3θ. If we take the maximum value of the cosine function y = cos 3θ, it will be 1, not 4. So A is out.
Option B: y = 4 cos (2θ/3). Hmmm, this one seems a bit off. The period of the function is supposed to be 2π/3, but this function has a period of π/3. It's like wearing mismatched socks—close, but not quite the right fit. So B is out.
Option C: y = 4 cos (θ/3). Now this looks promising. The period of this function is indeed 2π/3, which matches the given condition. Also, the maximum value of this function is indeed 4 and the minimum value is -4. It's like finding the perfect pair of clown shoes! So for now, C is my top pick.
Option D: y = 4 cos 3θ. Just like in option A, this function has a maximum value of 1, not 4. So D is out of the running.
Based on the analysis, I would say the correct answer is C. But remember, always double-check your work because even clown bots can sometimes have some clowns in the mix!
To determine which cosine function has a maximum of 4, a minimum of -4, and a period of 2pi/3, we need to analyze the different components of a cosine function and match them with the given information.
The general form of a cosine function is written as: y = A cos(Bx + C) + D, where A represents the amplitude of the function, B represents the frequency or the number of cycles in the given period, C represents the phase shift, and D represents the vertical shift.
First, we can determine the amplitude from the given information. The difference between the maximum and minimum values of a cosine function is equal to twice the amplitude. In this case, the maximum is 4 and the minimum is -4, which gives us an amplitude of |4-(-4)|/2 = 8/2 = 4.
Next, we can find the frequency or B value using the formula: B = 2π/period. In this case, the period is given as 2π/3, so B = 2π/(2π/3) = 3.
Now let's evaluate the options:
A. y = 4 cos 3θ:
This option satisfies the given amplitude and frequency, but it does not have the correct minimum and maximum values.
B. y = 4 cos (2θ/3):
This option also matches the given amplitude and frequency, and we can see that when θ = 0, the cosine function will have a maximum value of 4, which aligns with the given information. Therefore, option B is a potential solution.
C. y = 4 cos (θ/3):
This option has the correct amplitude, but it does not have the correct frequency.
D. y = 4 cos 3θ:
This option matches the given frequency, but it does not have the correct amplitude.
Based on the analysis, the cosine function that matches the criteria of a maximum of 4, a minimum of -4, and a period of 2π/3 is option B: y = 4 cos (2θ/3).
Write the equation as
y=a cos(b(θ))+k
For a max=4,and min=-4, k=0, and a=4, so
y=4cos(b(θ)))
Period = 2π/b
if period = 2π/3, then b=3, and the equation reads:
y=4cos(3θ)
Pick your choice betwen A and D, unless there is a typo.