Answer :
Sure, let's analyze and find the parameters for the general equation for each cosine function provided. The general form of a cosine function can be written as:
[tex]\[ y = A \cos(B(x - C)) \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude
- [tex]\( B \)[/tex] relates to the frequency
- [tex]\( C \)[/tex] is the phase shift
We will identify these parameters for each of the given cosine functions.
### For [tex]\( y = 4\pi \cos(3(x - \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is directly given by the coefficient of the cosine function, which is [tex]\(4\pi\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(3\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(\pi\)[/tex].
So the parameters for the first cosine function are:
- Amplitude [tex]\(A_1 = 12.566370614359172\)[/tex]
- Frequency [tex]\(B_1 = 3\)[/tex]
- Phase Shift [tex]\(C_1 = -3.141592653589793\)[/tex]
### For [tex]\( y = 3 \cos(4\pi(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(3\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(4\pi\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the second cosine function are:
- Amplitude [tex]\(A_2 = 3\)[/tex]
- Frequency [tex]\(B_2 = 12.566370614359172\)[/tex]
- Phase Shift [tex]\(C_2 = 3.141592653589793\)[/tex]
### For [tex]\( y = 3 \cos(0.5(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(3\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(0.5\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the third cosine function are:
- Amplitude [tex]\(A_3 = 3\)[/tex]
- Frequency [tex]\(B_3 = 0.5\)[/tex]
- Phase Shift [tex]\(C_3 = 3.141592653589793\)[/tex]
### For [tex]\( y = 4\pi \cos(0.2(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(4\pi\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(0.2\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the fourth cosine function are:
- Amplitude [tex]\(A_4 = 12.566370614359172\)[/tex]
- Frequency [tex]\(B_4 = 0.2\)[/tex]
- Phase Shift [tex]\(C_4 = 3.141592653589793\)[/tex]
In summary, the calculated parameters for each of the cosine functions are:
1. For [tex]\( y = 4\pi \cos(3(x - \pi)) \)[/tex]:
- Amplitude: 12.566370614359172
- Frequency: 3
- Phase Shift: -3.141592653589793
2. For [tex]\( y = 3 \cos(4\pi(x + \pi)) \)[/tex]:
- Amplitude: 3
- Frequency: 12.566370614359172
- Phase Shift: 3.141592653589793
3. For [tex]\( y = 3 \cos(0.5(x + \pi)) \)[/tex]:
- Amplitude: 3
- Frequency: 0.5
- Phase Shift: 3.141592653589793
4. For [tex]\( y = 4\pi \cos(0.2(x + \pi)) \)[/tex]:
- Amplitude: 12.566370614359172
- Frequency: 0.2
- Phase Shift: 3.141592653589793
[tex]\[ y = A \cos(B(x - C)) \][/tex]
where:
- [tex]\( A \)[/tex] is the amplitude
- [tex]\( B \)[/tex] relates to the frequency
- [tex]\( C \)[/tex] is the phase shift
We will identify these parameters for each of the given cosine functions.
### For [tex]\( y = 4\pi \cos(3(x - \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is directly given by the coefficient of the cosine function, which is [tex]\(4\pi\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(3\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(\pi\)[/tex].
So the parameters for the first cosine function are:
- Amplitude [tex]\(A_1 = 12.566370614359172\)[/tex]
- Frequency [tex]\(B_1 = 3\)[/tex]
- Phase Shift [tex]\(C_1 = -3.141592653589793\)[/tex]
### For [tex]\( y = 3 \cos(4\pi(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(3\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(4\pi\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the second cosine function are:
- Amplitude [tex]\(A_2 = 3\)[/tex]
- Frequency [tex]\(B_2 = 12.566370614359172\)[/tex]
- Phase Shift [tex]\(C_2 = 3.141592653589793\)[/tex]
### For [tex]\( y = 3 \cos(0.5(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(3\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(0.5\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the third cosine function are:
- Amplitude [tex]\(A_3 = 3\)[/tex]
- Frequency [tex]\(B_3 = 0.5\)[/tex]
- Phase Shift [tex]\(C_3 = 3.141592653589793\)[/tex]
### For [tex]\( y = 4\pi \cos(0.2(x + \pi)) \)[/tex]:
1. Amplitude ([tex]\(A\)[/tex]): The amplitude is given by the coefficient of the cosine function, which is [tex]\(4\pi\)[/tex].
2. Frequency ([tex]\(B\)[/tex]): The frequency is the coefficient of [tex]\(x\)[/tex] inside the cosine function, which is [tex]\(0.2\)[/tex].
3. Phase Shift ([tex]\(C\)[/tex]): The phase shift can be observed inside the cosine function. Here, it is [tex]\(-\pi\)[/tex].
So the parameters for the fourth cosine function are:
- Amplitude [tex]\(A_4 = 12.566370614359172\)[/tex]
- Frequency [tex]\(B_4 = 0.2\)[/tex]
- Phase Shift [tex]\(C_4 = 3.141592653589793\)[/tex]
In summary, the calculated parameters for each of the cosine functions are:
1. For [tex]\( y = 4\pi \cos(3(x - \pi)) \)[/tex]:
- Amplitude: 12.566370614359172
- Frequency: 3
- Phase Shift: -3.141592653589793
2. For [tex]\( y = 3 \cos(4\pi(x + \pi)) \)[/tex]:
- Amplitude: 3
- Frequency: 12.566370614359172
- Phase Shift: 3.141592653589793
3. For [tex]\( y = 3 \cos(0.5(x + \pi)) \)[/tex]:
- Amplitude: 3
- Frequency: 0.5
- Phase Shift: 3.141592653589793
4. For [tex]\( y = 4\pi \cos(0.2(x + \pi)) \)[/tex]:
- Amplitude: 12.566370614359172
- Frequency: 0.2
- Phase Shift: 3.141592653589793
