Analyze \(y~=~3\sin(2x)~+~1\). State \(A\), \(B\), \(h\), and \(D\); give the amplitude, reflection, period, phase shift, midline, range, and quarter-period; and list the \(\text{five}~\text{ordered}~\text{anchors}\) for one cycle with their direction or extrema.
What this problem is really about
Read the sine transformation by assigning each parameter one job: amplitude controls vertical distance from the midline, the inside coefficient controls period, and the shifts locate the cycle. Divide the period into four equal horizontal steps. A positive sine cycle then follows the ordered pattern midline rising, maximum, midline falling, minimum, and midline rising.