To determine the amplitude, period, and frequency of a trigonometric function, you need to understand the general forms of sine and cosine functions, which are:
y = A sin(Bx + C) + D and y = A cos(Bx + C) + D
In these equations:
- A represents the amplitude.
- B influences the period of the function.
- C is the phase shift, and D is the vertical shift.
Amplitude:
The amplitude of the function is the absolute value of A. It indicates how far the graph reaches above and below its midline (D). For instance, if your function is y = 3 sin(x), the amplitude is |3| = 3.
Period:
The period of the function is calculated using the formula Period = (2π) / |B|. This tells you how long it takes for one complete cycle of the wave to occur. If you have y = 2 cos(4x), the period is (2π) / |4| = π/2.
Frequency:
Frequency is the reciprocal of the period and shows how many cycles occur in a unit of time. It can be calculated with the formula Frequency = |B| / (2π). So, for the same function y = 2 cos(4x), the frequency would be |4| / (2π) = 2/π cycles per unit.
By following these steps, you can easily identify the amplitude, period, and frequency of any trigonometric function!