The function given is f(x) = 4 cos(2x) + 3. To determine the amplitude, period, and midline, let’s break it down step by step.
1. Amplitude
The amplitude of a cosine function is derived from the coefficient in front of the cosine term. In this case, the amplitude is the absolute value of the coefficient of cosine, which is:
Amplitude = |4| = 4
2. Period
The period of a cosine function can be calculated using the formula:
Period = rac{2 ext{π}}{b}
where b is the coefficient of x inside the cosine function. Here, b = 2, so:
Period = rac{2 ext{π}}{2} = ext{π}
3. Midline
The midline of a cosine function is determined by the vertical shift in the function, which is the constant added to the cosine term. In this function, the midline is given by:
Midline = 3
Summary
- Amplitude: 4
- Period: π
- Midline: 3
In conclusion, for the function f(x) = 4 cos(2x) + 3, the amplitude is 4, the period is π, and the midline is 3.