Identify the amplitude and period of the following functions.


p(t) = 2.5 sin ((1/2)(t-3))

Key Concepts

Amplitude

Amplitude refers to the maximum value of a periodic function, indicating how far the function's values deviate from its central axis. For sine functions, the amplitude is the coefficient in front of the sine term. In the function p(t) = 2.5 sin((1/2)(t-3)), the amplitude is 2.5, meaning the function oscillates between -2.5 and 2.5.

Period

The period of a function is the length of one complete cycle of the wave. For sine functions, the period can be calculated using the formula P = 2π / |b|, where b is the coefficient of t in the argument of the sine function. In p(t) = 2.5 sin((1/2)(t-3)), the coefficient b is 1/2, resulting in a period of 2π / (1/2) = 4π.

Phase Shift

Phase shift refers to the horizontal shift of a periodic function along the t-axis. It is determined by the constant added or subtracted inside the function's argument. In p(t) = 2.5 sin((1/2)(t-3)), the phase shift is 3 units to the right, as indicated by the (t-3) term, which affects where the function starts its cycle.