91Ó°ÊÓ

Plot key points on the graph of \(y = \sin x\). These key points occur when x takes on the values of \(x = 0\), \(x = \frac{\pi}{2}\), \(x = \pi\), \(x = \frac{3\pi}{2}\), and \(x = 2\pi\). We obtain the following coordinates: - (0, 0) - \(\left(\frac{\pi}{2}, 1\right)\) - \(\left(\pi, 0\right)\) - \(\left(\frac{3\pi}{2}, -1\right)\) - \(\left(2\pi, 0\right)\) These points are just one period of the sine function. Sketch a smooth curve passing through these points, and repeat the pattern to the left and right to complete the sine function graph.

To obtain the graph of \(y = 2\sin\frac{x}{2}\) from the graph of \(y = \sin x\), we need to apply the following transformations: 1. Vertical stretching by a factor of 2: This transformation will scale up the amplitude of the sine function from 1 to 2 (i.e., the sine function will oscillate between -2 and 2 instead of -1 and 1). 2. Horizontal stretching by a factor of 2: This transformation will scale the period of the sine function by a factor of 2. The period of the sine function is normally 2Ï€, but it will increase to 4Ï€ for the transformed function.

Now that we have identified the transformations needed, we can sketch the graph of \(y = 2\sin\frac{x}{2}\) based on the graph of \(y = \sin x\). Apply the transformations to the key points we used earlier for the sine function: - (0, 0) stays the same since all the transformations preserve the origin. - \(\left(\frac{\pi}{2}, 1\right)\) will be transformed to \(\left(\pi, 2\right)\): the x-coordinate is multiplied by 2 and the y-coordinate is doubled. - \(\left(\pi, 0\right)\) will be transformed to \(\left(2\pi, 0\right)\): the x-coordinate is multiplied by 2, while the y-coordinate remains the same. - \(\left(\frac{3\pi}{2}, -1\right)\) will be transformed to \(\left(3\pi, -2\right)\): the x-coordinate is multiplied by 2 and the y-coordinate is doubled. - \(\left(2\pi, 0\right)\) will be transformed to \(\left(4\pi, 0\right)\): the x-coordinate is multiplied by 2, while the y-coordinate remains the same. Plot these transformed points on the same set of axes as the graph of \(y = \sin x\). Sketch a smooth curve passing through these new points, extending the pattern to the left and right to complete the graph of \(y = 2\sin\frac{x}{2}\). Now you should have both the graph of \(y = \sin x\) and the graph of \(y = 2\sin\frac{x}{2}\) sketched on the same set of axes, and you can easily compare the two functions.