The function f(x) = 3 cos(4x) + 2 is a cosine function, which has specific characteristics that can be easily identified.
Amplitude
The amplitude of a cosine function is determined by the coefficient in front of the cosine term. In this case, the coefficient is 3. This means the amplitude is 3. The amplitude reflects the maximum distance the function reaches from its midline.
Period
The period of a cosine function can be calculated using the formula:
Period = (2π) / |B|
where B is the coefficient of x inside the cosine function. Here, B = 4. So, the period is:
Period = (2π) / 4 = π / 2.
Midline
The midline of the function is determined by the constant added to the cosine term, which in this case is +2. Therefore, the midline of the function is y = 2. This represents the horizontal line around which the cosine wave oscillates.
Summary
In summary, for the function f(x) = 3 cos(4x) + 2:
- Amplitude: 3
- Period: π / 2
- Midline: y = 2