To analyze the cosine function given by the equation y = 6cos(4x), we need to determine its domain, period, range, and amplitude.
Domain
The domain of a function is the set of all possible x-values for which the function is defined. For the cosine function, this is typically all real numbers. Therefore, the domain of y = 6cos(4x) is:
Domain: All real numbers, or (-∞, ∞)
Period
The period of a cosine function can be found using the formula Period = 2π / |b|, where b is the coefficient of x. In our equation, b = 4.
Thus, the period is:
Period = 2π / |4| = π/2
Range
The range of the cosine function is determined by its amplitude. The general form of a cosine function is y = Acos(Bx), where A is the amplitude. The amplitude of this function is 6, meaning the function oscillates between -6 and 6.
Therefore, the range of y = 6cos(4x) is:
Range: [-6, 6]
Amplitude
The amplitude of a cosine function is the absolute value of A. In this case, the amplitude is:
Amplitude: 6
Summary
- Domain: All real numbers (-∞, ∞)
- Period: π/2
- Range: [-6, 6]
- Amplitude: 6