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Chapter 4: Problem 47

Sketch the graph of the function. (Include two full periods.) $$y=-\sin \frac{2 \pi x}{3}$$

Short Answer

Expert verified
The graph of the function \(y=-\sin \frac{2 \pi x}{3}\) is a sine curve reflected over the x-axis with a period of 3. It starts from the origin, reaches a minimum value of -1 at \(\frac{3}{4}\), returns to 0 at \(\frac{3}{2}\), reaches a maximum of 1 at \(\frac{9}{4}\), and ends up at 0 at x=3. The same pattern is repeated for the next period.

Step by step solution

01

Analyzing the amplitude and period of the function

To begin with, the amplitude of the function is given by the absolute value of the coefficient of the sine function. Here it is 1, but there is a negative sign, indicating that graph will be reflected over the x-axis (or simply 'inverted'). The period of the function will be \(\frac{2\pi}{\mid\frac{2\pi}{3}\mid}=3\). So, the graph will complete a full cycle in the interval from 0 to 3 (instead of 0 to \(2\pi\) as for standard sine).
02

Plotting the key points

Now the main points for the sine function within one period [0, 3] are plotted. That's 0, \(\frac{3}{4}\), \(\frac{3}{2}\), \(\frac{9}{4}\) and 3. As the function is reflected over the x-axis, the values at these points are 0, -1, 0, 1, 0.
03

Sketching the graph

After plotting these key points, a smooth curve is drawn through them to represent the function. As the sine function is periodic, the same pattern is repeated for the next period. Thus, a complete sketch of the function \(y=-\sin \frac{2 \pi x}{3}\) over two periods is completed.

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