What is the amplitude and the period of the function \( y=3-2 \cos 3 x \) ? a \( 3, \frac{2 \pi}{3} \) b \( 5, \frac{2 \pi}{3} \) c \( 1, \frac{\pi}{3} \) d \( 2, \frac{2 \pi}{3} \) Check It

The amplitude of a cosine function is given by the absolute value of the coefficient before the cosine term. In the function \( y=3-2 \cos 3x \), the coefficient is \(-2\), so the amplitude is \(|-2| = 2\). The period is determined by the formula \(\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\). Here, \(B = 3\), which means the period is \(\frac{2\pi}{3}\). Therefore, the correct answer is **d \( 2, \frac{2 \pi}{3} \)**. Understanding the function better, the graphical representation would show that the curve oscillates between the maximum value of \(3-(-2) = 5\) and the minimum value of \(3-2 = 1\). Thus, your amplitude indicates how far the graph stretches from its midline at \(y = 3\). For further exploration, you can dive into trigonometric identities or explore how different transformations impact the graphs of cosine functions. This can include changing phase shifts or experimenting with vertical shifts to see how they affect the overall appearance of your function!