The cosine function is a periodic function that oscillates between -1 and 1. To analyze the specific function given, y = 3cos(4x), we can break it down into its components: amplitude, period, and range.
Amplitude
The amplitude of a cosine function y = A * cos(Bx) is determined by the coefficient A. In this case, the amplitude is:
- Amplitude = |A| = |3| = 3
This means the function will reach a maximum value of 3 and a minimum value of -3.
Period
The period of a cosine function is calculated using the formula:
Period = (2π) / |B|
For our function, B = 4, so the period is:
- Period = (2π) / |4| = π / 2
This means the function completes one full cycle every π/2 units along the x-axis.
Range
The range of the cosine function defines the values that y can take. Since we know the amplitude is 3, the range for the function y = 3cos(4x) can be determined as:
- Range: [-3, 3]
This indicates that the values of y will vary from -3 to 3.
Summary
To summarize:
- Amplitude: 3
- Period: π/2
- Range: [-3, 3]
This gives you a complete understanding of the behavior of the cosine function y = 3cos(4x). Happy studying!