Problem 103 - Algebra and Trigonometry

Question

Use a graphing utility to graph two periods of the function. $$y=0.2 \sin \left(\frac{\pi}{10} x+\pi\right)$$

Step-by-Step Solution

Answer

The graph will show two complete cycles of a sinusoidal function with amplitude of 0.2, a period of 20, and shifted left by \(\pi\). The x-range is \[-\pi, 40 - \pi\] and the y-range is \[-0.2, 0.2\].

Step 1: Identify the attributes of the sinusoidal function

From the equation \(y = 0.2 \sin\left(\frac{\pi}{10} x + \pi\right)\), we can identify the attributes of the function. The amplitude is 0.2, which determines the maximum and minimum values of the function. The period is given by \( \frac{2\pi}{B}\), and B in this equation is \( \frac{\pi}{10}\), so the period is 20. The function is shifted left by \(\pi\).

Step 2: Set the range for x

Given the period is 20, and we need to graph two periods, the x-range will be 40 units. Because the function is shifted left by \(\pi\), we can start the x-range at \(-\pi\) and end at \(40 - \pi\). Thus, the x-range becomes \[-\pi, 40 - \pi\).

Step 3: Set the y-range

The y-values are determined by the amplitude of the function. The maximum y-value occurs at the peak of each wave, which is equal to the amplitude. The minimum y-value occurs at the trough of each wave, which is equal to the negative of the amplitude. Hence, the y-range will be \[-0.2, 0.2\].

Step 4: Use a graphing utility to graph the function

Input the equation into a graphing utility, using the determined x and y ranges. The x and y scales should be set so that the key points (maximum, minimum, and zeros) can be clearly seen. The graph should show two complete waves of the sine function, properly shifted and with the appropriate amplitude.