What are the Amplitude, Period, and Midline of f(x) = 5 sin(x − π) + 3?
We will be using the phase shift standard form to solve this.
Answer: The Amplitude, Period, and Midline of f(x) = 5 sin(x − π) + 3 are 5, 2π, and y = 3 respectively.
Let's solve this step by step.
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