2.Graph f(x) = 1/4 cos(1/2 x + pi/2). Find the amplitude, period, and phase shift of the function.

3. Graph f(x) = -cos(3x + pi). Find the amplitude, period, and phase shift of the function.

To find the amplitude, period, and phase shift of a trigonometric function, we need to understand the general form of the function and how it relates to these properties.

1. For the function f(x) = A * cos(Bx + C), the amplitude is given by |A|, the period is given by 2π/|B|, and the phase shift is given by -C/B.

2. For the function f(x) = -cos(Bx + C), the amplitude is 1, the period is 2π/|B|, and the phase shift is -C/B.

Now, let's apply these formulas to your given functions:

2. f(x) = 1/4 cos(1/2 x + π/2):
Amplitude: The coefficient of cos function is 1/4, so the amplitude is |1/4| = 1/4.

Period: The coefficient of x in the argument of the cos function is 1/2, so the period is 2π/|1/2| = 2π/(1/2) = 4π.

Phase Shift: The constant term is π/2, so the phase shift is -π/2 / (1/2) = -π/2 * 2/1 = -π.

Therefore, the amplitude, period, and phase shift of the function f(x) = 1/4 cos(1/2 x + π/2) are 1/4, 4π, and -π, respectively.

3. f(x) = -cos(3x + π):
Amplitude: The coefficient of cos function is -1, so the amplitude is 1.

Period: The coefficient of x in the argument of the cos function is 3, so the period is 2π/|3| = 2π/3.

Phase Shift: The constant term is π, so the phase shift is -π / 3 = -π/3.

Therefore, the amplitude, period, and phase shift of the function f(x) = -cos(3x + π) are 1, 2π/3, and -π/3, respectively.