Which of the following is a cosine equation having an amplitude of 2 and period of 6?
Question
Which of the following is a cosine equation having an amplitude of 2 and period of 6?
Solution 1
The cosine function is generally given by the equation y = A cos(B(x - C)) + D, where:
- A is the amplitude
- B determines the period
- C is the phase shift
- D is the vertical shift
Given that the amplitude is 2 and the period is 6, we can substitute these values into the equation. The amplitude is the absolute value of A, so A = 2.
The period of the cosine function is given by (2π) / |B|. So, if the period is 6, we can set up the equation (2π) / |B| = 6. Solving for B, we get B = (2π) / 6 = π/3.
Therefore, the cosine equation with an amplitude of 2 and a period of 6 is y = 2 cos((π/3)x).
Solution 2
The cosine function is generally given by the equation y = A cos(B(x - C)) + D, where:
- A is the amplitude
- B determines the period
- C is the phase shift
- D is the vertical shift
Given that the amplitude is 2 and the period is 6, we can substitute these values into the equation. The amplitude is the absolute value of A, so A = 2. The period of the cosine function is given by (2π) / |B|. So, if the period is 6, then B = (2π) / 6 = π/3.
Therefore, the cosine equation with an amplitude of 2 and a period of 6 is y = 2 cos((π/3)x).
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