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

Question

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

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

Answer 1

amplitude is 1/4 given

when x/2 = 2pi we have a period
4 pi is period

- pi/2 is phase shift

same deal for the next one

Answer 2

To find the amplitude, period, and phase shift of the given functions, we need to understand their general form:

1. f(x) = A cos(Bx + C)

Here, A represents the amplitude, B determines the period, and C indicates the phase shift.

Let's solve each problem one by one:

2. f(x) = (1/4) cos((1/2)x + π/2)

For this function:
- The amplitude, A, is 1/4. The amplitude is the value that determines the maximum vertical distance from the middle line of the graph (the average value of the function).
- The period, T, is given by the formula 2π/B, where B is the coefficient of x. In this case, B = 1/2. Thus, the period is T = 2π/(1/2) = 4π. So, the function repeats itself every 4π units.
- The phase shift, C, is given by -C/B, where B is the coefficient of x. In this case, B = 1/2 and C = π/2. Therefore, the phase shift is -π/2 ÷ (1/2) = -π. The phase shift determines how much the graph has shifted horizontally.

So, for f(x) = (1/4) cos((1/2)x + π/2), the amplitude is 1/4, the period is 4π, and the phase shift is -π.

3. f(x) = -cos(3x + π)

For this function:
- The amplitude, A, is 1. The negative sign does not affect the amplitude.
- The period, T, is given by the formula 2π/B, where B is the coefficient of x. In this case, B = 3. Thus, the period is T = 2π/3. So, the function repeats itself every 2π/3 units.
- The phase shift, C, is given by -C/B, where B is the coefficient of x. In this case, B = 3 and C = π. Therefore, the phase shift is -π ÷ 3.

So, for f(x) = -cos(3x + π), the amplitude is 1, the period is 2π/3, and the phase shift is -π/3.

Remember that the amplitude represents the maximum vertical distance from the middle line, the period indicates how often the function repeats itself, and the phase shift shows the horizontal shift of the graph.