The cosine function has a specific structure that allows us to determine its characteristics easily. Let’s analyze the function y = 6cos(4x).
1. Domain
The domain of a cosine function is all real numbers since the function is defined for every value of x. Therefore, for y = 6cos(4x), the domain is:
- Domain:
(−∞, +∞)
2. Period
The period of a cosine function is determined by the coefficient of x within the cosine. For the function cos(kx), the period is given by the formula:
- Period =
2π / |k|
In our function, k = 4.
Thus, we calculate the period as:
- Period =
2π / 4 = π/2
3. Amplitude
The amplitude of the cosine function is the absolute value of the coefficient in front of the cosine, which indicates how far the function stretches from its midline.
- Amplitude:
|6| = 6
4. Range
The range of a cosine function is typically between -1 and 1. However, with the coefficient of 6 in front, the range will stretch out to:
- Range:
[-6, 6]
Summary
To summarize:
- Domain:
(−∞, +∞) - Period:
π/2 - Amplitude:
6 - Range:
[-6, 6]
Understanding these characteristics will help you work with the cosine function in various mathematical applications!