What is the amplitude and the period of the function \( y=2-4 \sin (2 \pi-6 x) \) ? a Amplitude is 3 and period is \( \frac{\pi}{3} \) b Amplitude is 4 and period is \( \frac{\pi}{3} \) c Amplitude is 4 and period is \( \frac{\pi}{6} \) d Amplitude is 8 and period is \( \frac{2 \pi}{3} \) Check it

The function \( y = 2 - 4 \sin(2\pi - 6x) \) can be analyzed to find the amplitude and period. The amplitude of the sine function is determined by the coefficient of sine, which is 4. The period can be calculated using the formula \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \(x\) in the argument of the sine function. Here, \( b = -6 \), so the period is \( \frac{2\pi}{6} = \frac{\pi}{3} \). Thus, the amplitude is 4 and the period is \( \frac{\pi}{3} \). So the correct answer is: b Amplitude is 4 and period is \( \frac{\pi}{3} \). To dive deeper, the sine function's amplitude reflects the height of its peaks relative to its midline. So, when you see a coefficient before the sine, that's the maximum offset from this midline. The period tells us how often the wave repeats; shorter periods result in more oscillations in the same space. This is crucial in many fields, such as engineering and physics, where wave behavior is fundamental.