What is the amplitude, period, and midline of f(x) = 5 sin(x - π) + 3?

Solution:

Given that, f(x) = 5 sin(x − π) + 3

We have a standard form for phase shift:

f(x) = a sin(bx + c) + d

Here, a = 5, b = 1, c = -π, d = 3

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

On Comparing we get: Amplitude = 5, Time Period = 2π, Phase Shift = -π, Vertical Shift = 3

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) + 2 has a midline of y = 2.

⇒ The midline of function f(x) = 5 sin(x − π) + 3 is y = 3

Hence, the amplitude, period, and midline of f(x) = 5 sin(x − π) + 3 are 5, 2π, and y = 3 respectively.

What is the amplitude, period, and midline of f(x) = 5 sin(x - π) + 3?

Summary:

The amplitude, period, and midline of f(x) = 5 sin(x - π) + 3 are 5, \(2\pi \) and 3.