Understanding Amplitude, Period, and Phase Shift
The function you provided is f(x) = 3 cos(4x) + 6, which is a transformation of the basic cosine function. Let’s break it down step-by-step to identify its amplitude, period, and phase shift.
1. Amplitude
Amplitude in a cosine function describes the height of the wave from its center line to its peak. In the equation f(x) = A cos(Bx) + D, A represents the amplitude.
For our function:
- A = 3
Thus, the amplitude is 3. This means the wave will oscillate 3 units above and below its center line.
2. Period
The period of a cosine function is determined by the B value in the equation, which affects how frequently the wave repeats. The formula for calculating the period is:
Period = (2π) / |B|
In our case:
- B = 4
Plugging in the value:
Period = (2π) / |4| = π/2
Thus, the period of the function is π/2, indicating that the wave completes one full cycle in that interval.
3. Phase Shift
Phase shift refers to a horizontal shift in the graph of the function. If there’s a horizontal translation, it will typically be represented in the function as f(x) = A cos(B(x – C)) + D, where C indicates the phase shift. In your function f(x) = 3 cos(4x) + 6, there’s no additional term inside the cosine function that would shift it to the left or right.
This means the phase shift is:
- Phase Shift = 0
So, the graph of the cosine function has no horizontal translation and begins at the origin of the cosine function.
Conclusion
In summary, for the function f(x) = 3 cos(4x) + 6:
- Amplitude: 3
- Period: π/2
- Phase Shift: 0
Understanding these components helps to visualize and analyze the behavior of the cosine function as it oscillates!