To find the amplitude, period, and phase shift of the function f(x) = 3 cos(4x) + 6, we can analyze the components of the cosine function.
Amplitude
The amplitude of a cosine function is determined by the coefficient in front of the cosine term. In this case, the amplitude is 3. This means that the graph of the function oscillates 3 units above and below its midline.
Period
The period of a cosine function is calculated using the formula Period = rac{2 ext{π}}{B}, where B is the coefficient of x inside the cosine function. Here, B = 4. Thus, the period is:
Period = rac{2π}{4} = rac{π}{2}
This means that the function completes one full cycle every π/2 units along the x-axis.
Phase Shift
The phase shift for any cosine function of the form f(x) = A cos(B(x – C)) + D is given by C. In this function, however, there is no horizontal shift since the cosine is simply cos(4x) without any addition or subtraction inside the argument. Thus, the phase shift is 0.
Conclusion
In summary, for the function f(x) = 3 cos(4x) + 6, we have:
- Amplitude: 3
- Period: π/2
- Phase Shift: 0