A certain cosine function completes 6 cycles over the interval [0,2π].
Which function rule could model this situation?
f(x)=cos(1/6x)
f(x)=cos(x)+6
f(x)=cos(6x)
f(x)=6cos(x)
Yes
so the third one?
6 cycles is 6 * 2π radians
that is 6 x
Well, since the function completes 6 cycles over the interval [0,2π], we know that it oscillates 6 times. So, we need a coefficient in front of the x that divides it in such a way that when x = 2π, we have completed 6 cycles.
The correct function rule that models this situation is f(x) = cos(6x).
You see, by multiplying x by 6, we essentially "speed up" the graph, causing it to complete 6 cycles within the interval [0,2π]. So, grab your calculator and calculate some cosine values with this function! It's quite the "cos-perience"!
To find the correct function rule that models a cosine function completing 6 cycles over the interval [0,2π], we need to understand the relationship between the coefficient of x and the number of cycles.
In a cosine function, the coefficient of x determines the frequency and affects the number of cycles. The general form of a cosine function is f(x) = A*cos(Bx), where A represents the amplitude and B represents the frequency or the number of cycles.
In this case, the given function completes 6 cycles over the interval [0,2π].
To find the frequency B, we can use the formula B = 2π/period, where the period is the interval that completes one full cycle.
In this case, the given function completes 6 cycles over the interval [0,2π], so the period is (2π)/6 = 2π/6 = π/3.
Therefore, the frequency B is given by B = 2π/(π/3) = 2π*(3/π) = 6.
Now, let's take a look at the function rules provided and determine which one represents the correct model:
1. f(x) = cos(1/6x): This function has a frequency of 1/6, which means it completes only 1/6 of a cycle over the interval [0,2π]. This is not the correct model.
2. f(x) = cos(x) + 6: This function has a constant term of +6, which would shift the graph of the cosine function upward by 6 units. It does not affect the number of cycles. This is not the correct model.
3. f(x) = cos(6x): This function has a frequency of 6, which means it completes 6 cycles over the interval [0,2π]. This matches the given situation, so this is the correct model.
4. f(x) = 6*cos(x): This function has an amplitude of 6, which would stretch the graph of the cosine function vertically by a factor of 6. It does not affect the number of cycles. This is not the correct model.
Therefore, the correct function rule that models the given situation is f(x) = cos(6x).