find the amplitude, period, and phase shift of the function. graph the function. be sure to label key…

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is (y = A\sin(Bx - C)+D). For the function (y = 5\sin(4x-\pi)), we have (A = 5), (B = 4), (C=\pi), (D = 0).

Step2: Calculate the amplitude

The amplitude is given by (|A|). So, (|A|=|5| = 5).

Step3: Calculate the period

The period of a sine function (y = A\sin(Bx - C)+D) is (T=\frac{2\pi}{|B|}). Here, (B = 4), so (T=\frac{2\pi}{4}=\frac{\pi}{2}).

Step4: Calculate the phase - shift

The phase - shift is given by (\frac{C}{B}). Since (C=\pi) and (B = 4), the phase - shift is (\frac{\pi}{4}).

For graphing, we can find key points. Let (u=4x-\pi). When (u = 0) (corresponds to the start of a cycle after phase - shift), (4x-\pi=0\Rightarrow x=\frac{\pi}{4}). When (u=\frac{\pi}{2}), (4x-\pi=\frac{\pi}{2}\Rightarrow x=\frac{3\pi}{8}), and (y = 5). When (u=\pi), (4x-\pi=\pi\Rightarrow x=\frac{\pi}{2}), and (y = 0). When (u=\frac{3\pi}{2}), (4x-\pi=\frac{3\pi}{2}\Rightarrow x=\frac{5\pi}{8}), and (y=-5). When (u = 2\pi), (4x-\pi=2\pi\Rightarrow x=\frac{3\pi}{4}), and (y = 0).

Answer:

Amplitude: (5), Period: (\frac{\pi}{2}), Phase - shift: (\frac{\pi}{4})