State the amplitude, period, phase shift, and vertical shift of f(x)=-1/4sin(6x+2pi)+9

To determine the amplitude, period, phase shift, and vertical shift of the function f(x) = -1/4sin(6x+2π) + 9, let's break it down step by step:

1. Amplitude: The amplitude of a trigonometric function represents the maximum value it reaches from its midline. In this case, the coefficient in front of the sine function is -1/4. The absolute value of this coefficient, 1/4, gives us the amplitude. Therefore, the amplitude of f(x) is 1/4.

2. Period: The period of a trigonometric function is the distance between two consecutive points on the graph that have identical values. For the sine function, the general formula for the period is 2π divided by the coefficient in front of the x-axis (6 in this case). So, the period of f(x) is 2π/6, which simplifies to π/3.

3. Phase Shift: The phase shift occurs when the entire graph of a function is shifted horizontally either to the left or to the right. To determine the phase shift, we need to set the expression inside the sine function, 6x+2π, equal to zero and solve for x. In this case, we have:

6x + 2π = 0
6x = -2π
x = -2π/6

Therefore, the phase shift of f(x) is -2π/6, which simplifies to -π/3.

4. Vertical Shift: The vertical shift, also known as the vertical displacement, refers to any shift of the entire graph of a function up or down. In this case, the function is vertically shifted up by 9 units because of the "+9" term at the end. Thus, the vertical shift of f(x) is 9 units.

To summarize:
- Amplitude: 1/4
- Period: π/3
- Phase Shift: -π/3
- Vertical Shift: 9 units