What are the Amplitude, Period, Phase Shift, and Midline of f(x) = 2 sin(x + π) − 4?

We will be using the phase shift standard form to solve this.

Answer: The Amplitude, Period, Phase Shift, and Midline of f(x) = 2 sin(x + π) − 4 are 2, 2π, π, and y = -4 respectively.

Let's solve this step by step.

Explanation:

Given that, f(x) = 2 sin(x + π) − 4

We have a standard form for phase shift:

f(x) = a sin(bx + c) + d

Here, a = 2, b = 1, c = π, d = -4

Where, Amplitude = a, Time Period = 2π/b, Phase Shift = c, Vertical Shift = d

On Comparing we get: Amplitude = 2, Time Period = 2π, Phase Shift = π, Vertical Shift = -4

The midline is parallel to the x-axis and runs between the maximum and minimum value (i.e., amplitudes)

For the function f(x) = sinx, midline is y = 0, midline is affected by any vertical shift/translations. For example, y = sin(x + π) + 4 has a midline of y = 4.

⇒ The midline of function f(x) = 2 sin(x + π) − 4 is y = -4

Hence, the amplitude, period, phase shift, and midline of f(x) = 2 sin(x + π) − 4 are 2, 2π, π, and y = -4 respectively.