To analyze the function f(x) = 4 cos(2x) + 3, we can identify its amplitude, period, and midline based on its form.
Amplitude
The amplitude of a cosine function is the coefficient in front of the cosine term. In this case, the coefficient is 4. This means the amplitude is 4. Amplitude represents the maximum deviation from the midline of the wave.
Period
The period of a cosine function defined as f(x) = A cos(Bx) + D is calculated using the formula: Period = (2π) / |B|. For our function, B = 2, so:
Period = (2π) / |2| = π. This indicates that the function completes one full cycle in π units along the x-axis.
Midline
The midline of the function is determined by the vertical shift represented by D in the function. Here, D = 3, which means the midline is the horizontal line y = 3. This line divides the maximum and minimum values of the waveform.
Summary
- Amplitude: 4
- Period: π
- Midline: y = 3
In conclusion, the function has an amplitude of 4, a period of π, and a midline at y = 3.