Question

amplitude =
equation (6) =
period =
phase shift =
(in terms of the sine function)

Explanation:

Step1: Find the amplitude

The amplitude of a sine - wave \(y = A\sin(Bx - C)+D\) is given by \(|A|\). From the graph, the maximum value \(y_{max}=3\) and the minimum value \(y_{min}=- 3\). The amplitude \(A=\frac{y_{max}-y_{min}}{2}=\frac{3 - (-3)}{2}=3\).

Step2: Find the period

The general formula for the period of a sine function \(y = A\sin(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). From the graph, we can see that the length of one full cycle (period) \(T = 4\).

Step3: Find the phase - shift

The general form of a sine function is \(y = A\sin(B(x - h))+D\), where \(h\) is the phase - shift. For a sine function \(y = A\sin(Bx - C)+D\), the phase - shift is \(h=\frac{C}{B}\). Since the graph passes through the origin \((0,0)\) and there is no horizontal shift (the standard sine function \(y=\sin(x)\) passes through \((0,0)\) after considering amplitude and period), the phase - shift is \(0\).

Step4: Write the equation

Using the form \(y = A\sin(Bx)+D\), where \(A = 3\), \(B=\frac{2\pi}{T}=\frac{2\pi}{4}=\frac{\pi}{2}\), and \(D = 0\). The equation is \(y = 3\sin(\frac{\pi}{2}x)\)

Answer:

Amplitude \(=3\); Period \(=4\); Phase Shift \(=0\); Equation \(y = 3\sin(\frac{\pi}{2}x)\)