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