Question

1. sketch the graph of the equation $y=-2\sin(\pi(x - 1))-4$.

Explanation:

Step1: Identify the general form of the sine function

The general form of a sine function is \(y = A\sin(B(x - C))+D\). For the given function \(y=-2\sin(\pi(x - 1))-4\), we have \(A=-2\), \(B = \pi\), \(C = 1\), \(D=-4\).

Step2: Determine the amplitude

The amplitude is \(|A|\). Here, \(|A|=|-2| = 2\).

Step3: Calculate the period

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

Step4: Find the phase - shift

The phase - shift is \(C\). Here, the phase - shift is \(1\) (to the right).

Step5: Determine the vertical - shift

The vertical - shift is \(D=-4\).

To sketch the graph:

• Start with the basic sine function \(y = \sin(x)\) which has a period of \(2\pi\), amplitude \(1\), no phase - shift and no vertical - shift.
• Compress the period of \(y = \sin(x)\) from \(2\pi\) to \(2\) (because \(B=\pi\)).
• Reflect the graph about the \(x\) - axis (because \(A=-2\)).
• Stretch the graph vertically by a factor of \(2\) (because \(|A| = 2\)).
• Shift the graph \(1\) unit to the right (because \(C = 1\)).
• Shift the graph \(4\) units down (because \(D=-4\)).

Answer:

The graph of \(y=-2\sin(\pi(x - 1))-4\) has amplitude \(2\), period \(2\), phase - shift \(1\) unit to the right and vertical - shift \(4\) units down.