Identify the amplitude and period of \( g(x)=\frac{1}{2} \cos 4 \pi x \). Then identify the graph of the function and describe the graph of \( g \) as a transformation of the graph of its parent function. The amplitude is \( \square \) and the period is \( \square \).

The amplitude of the function \( g(x) = \frac{1}{2} \cos 4\pi x \) is \( \frac{1}{2} \), which indicates how far the graph stretches above and below its centerline. The period, determined by the formula \( \frac{2\pi}{b} \) where \( b \) is the coefficient of \( x \), is \( \frac{2\pi}{4\pi} = \frac{1}{2} \). When graphing \( g(x) \), you’ll see it as a transformation of the parent function \( \cos x \): the original cosine wave has been vertically compressed by a factor of \( \frac{1}{2} \) and horizontally stretched to have a period of \( \frac{1}{2} \). Essentially, while the classic cosine graph oscillates between -1 and 1, your graph toggles between -0.5 and 0.5 over a shortened cycle!