To analyze the function f(x) = 3 cos(4x) + 2, we can break it down into its characteristics: amplitude, period, and midline.
1. Amplitude
The amplitude of a cosine function can be determined from its coefficient in front of the cosine. In this case, the coefficient is 3. The amplitude is the distance from the midline to the maximum or minimum value of the function. Therefore, the amplitude is:
Amplitude = 3
2. 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. In this function, b = 4.
Substituting in the value of b gives us:
Period = (2π) / 4 = π / 2
3. Midline
The midline of a trigonometric function is the horizontal line that runs through the middle of the function’s graph. It can be found by identifying the constant term added or subtracted from the cosine function. In this case, the function includes a +2, so the midline is:
Midline = 2
Summary
For the function f(x) = 3 cos(4x) + 2, we summarize:
- Amplitude: 3
- Period: π / 2
- Midline: 2