Write an equation of a cosine function with Amplitude=4, Period=pi/2, and Phase Shift=-pi? Also write an equation of a cosine function with Amplitude=1/2, Period=6pi, and Phase Shift=pi/3?
To find the equation of a cosine function with the given parameters, you can use the general form of a cosine function:
y = A * cos(B(x - C)) + D,
where:
A is the amplitude,
B is the frequency (which is determined by the period),
C is the phase shift, and
D is the vertical shift.
Let's start with the first equation:
Amplitude (A) = 4,
Period = π/2,
Phase Shift (C) = -π.
The amplitude tells us that the maximum and minimum values of the cosine function will be 4 and -4, respectively.
The period is the distance covered by one complete cycle of the function. In this case, it is π/2, which is equivalent to 90 degrees (or a quarter of a cycle).
The phase shift determines the horizontal shift of the function. In our case, it is -π, which means the function is shifted to the right by π units.
Since the general form of the cosine function has a vertical shift (D) of 0, we do not need to consider it here.
Putting it all together, we have:
y = 4 * cos(2(x + π/4))
Now let's move on to the second equation:
Amplitude (A) = 1/2,
Period = 6π,
Phase Shift (C) = π/3.
The amplitude tells us that the maximum and minimum values of the cosine function will be 1/2 and -1/2, respectively.
The period is 6π, which means it covers six complete cycles in a length of 2π. Therefore, in each complete cycle, the function covers a distance of 2π/6 = π/3.
The phase shift is π/3, indicating that the function is shifted to the right by 1/3 of a cycle.
Again, since the vertical shift (D) is not mentioned, we do not need to consider it.
Putting it all together, we have:
y = (1/2) * cos(12(x - π/3))
These equations represent cosine functions with the given parameters.