To graph the functions y = 2 + sin(x) and y = 2sin(x), it's essential to understand how transformations affect the sine wave. Adding 2 shifts the entire graph upward by 2 units, while multiplying by 2 changes the amplitude, making the peaks and troughs twice as high and low. The discussion clarifies that the y-axis remains fixed, and the sine wave's basic format is f(θ) = Asin(θ), where A represents the amplitude. The transformation insights help visualize the sine function's behavior with these modifications. Understanding these principles is crucial for accurately graphing trigonometric functions.