Short Answer 
The sine function has a period of 4œA, amplitude of -3, and midline at y = 2, leading to the function y = -3 sin(1/2 x) + 2. To graph it, plot the midline and the maximum and minimum points, reflecting over the midline due to the negative amplitude.
Step 1: Understand the Key Features
To analyze the sine function, first note its key features. This function has a period of 4œA, an amplitude of 3, and a midline at y = 2. The y-intercept is (0, 2), indicating where the graph crosses the y-axis, which confirms that the function oscillates around the midline. Additionally, the function is a reflection of its parent function because of the negative amplitude.
Step 2: Determine the Sine Function Formula
Use the general formula for a sine function, which is represented as y = A sin(Bx) + C. In this formula: A is the amplitude, B is related to the period, and C is the midline. Substitute the known values: A = -3 (due to reflection), C = 2, and calculate B using B = 2œA/Period. Hence, B = 1/2, leading to the function y = -3 sin(1/2 x) + 2.
Step 3: Plotting the Graph
To plot the graph, begin by marking the midline at (0, 2). From this midline, determine the high and low points based on the amplitude: the maximum point is found at (-œA, 5) and the minimum point at (œA, -1). Create a table of values to plot additional points if necessary, and draw a smooth curve to represent the sine wave, ensuring to reflect over the midline as indicated by the negative amplitude.