Question

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\sin x, \quad y=2 \sin \frac{x}{2}\)

Short answer

On the same set of axes, sketch the graph of \(y=\sin x\) with key points (0, 0), \(\left(\frac{\pi}{2}, 1\right)\), \(\left(\pi, 0\right)\), \(\left(\frac{3\pi}{2}, -1\right)\), and \(\left(2\pi, 0\right)\). Apply vertical stretching and horizontal stretching transformations to obtain the graph of \(y=2\sin\frac{x}{2}\) with key points (0, 0), \(\left(\pi, 2\right)\), \(\left(2\pi, 0\right)\), \(\left(3\pi, -2\right)\), and \(\left(4\pi, 0\right)\).

Step-by-step solution

Sketch the graph of \(y=\sin x\)

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.

Identify the transformations required

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 2π, but it will increase to 4π for the transformed function.

Sketch the graph of \(y=2 \sin \frac{x}{2}\) after applying transformations

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.