Identify the period and amplitude of each function.
y=3cosÈ
y=-2cos2t
y=2cos1/2t
To identify the period and amplitude of each function, we need to understand the properties of cosine functions.
The general equation for a cosine function is y = A * cos(Bx), where A represents the amplitude and B represents the frequency or angular frequency. In this case, we have y = 3cosÈ, y = -2cos2t, and y = 2cos(1/2)t.
1. For y = 3cosÈ:
- The amplitude (A) is 3, which represents the maximum distance from the centerline to the peak or trough of the function.
- Since there is no Bx term in the equation, the value of B is 1.
- The period (T) can be calculated using the formula T = 2π/B, where B is the coefficient of x. In this case, B = 1, so the period is 2π.
Therefore, for y = 3cosÈ, the amplitude is 3, and the period is 2π.
2. For y = -2cos2t:
- The amplitude (A) is 2, which represents the maximum distance from the centerline to the peak or trough of the function.
- The coefficient of t (B) is 2, which determines the frequency or angular frequency.
- The period (T) can be calculated using the formula T = 2π/B. In this case, B = 2, so the period is π.
Therefore, for y = -2cos2t, the amplitude is 2, and the period is π.
3. For y = 2cos(1/2)t:
- The amplitude (A) is 2, which represents the maximum distance from the centerline to the peak or trough of the function.
- The coefficient of t (B) is 1/2, which determines the frequency or angular frequency.
- The period (T) can be calculated using the formula T = 2π/B. In this case, B = 1/2, so the period is 4π.
Therefore, for y = 2cos(1/2)t, the amplitude is 2, and the period is 4π.
To summarize:
- For y = 3cosÈ: amplitude = 3, period = 2π.
- For y = -2cos2t: amplitude = 2, period = π.
- For y = 2cos(1/2)t: amplitude = 2, period = 4π.