Problem 14 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-2 \sin \pi x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y= -2 \sin \pi x\) is 2 and the period is 2.

Step 1: Amplitude Calculation

From the given function \(y= -2 \sin \pi x\), the amplitude is the absolute value of the coefficient of the sin function. That means amplitude = \(|-2|\) = 2.

Step 2: Period Calculation

The period is the reciprocal of the absolute value of the coefficient of x inside the sin function. In the given function, \( \pi \) is the coefficient of x. Therefore, period = \( \frac{2\pi}{|\pi|} \) which equals to 2.

Step 3: Graphing the Function

When graphing, the period indicates the complete cycle of one full wave. Start by marking the period on the x-axis and then mark the mid point and quarter points. The function starts from origin, the maximum point is the mid of the first quarter, the second quarter ends at origin, third quarter mid point is the minimum and it again ends at origin on reaching the period value. Since the amplitude given is 2, the maximum and minimum points on the y-axis will be +2 and -2 respectively. Draw the curve joining these points to represent one full wave of the sinusoidal function \(y= -2 \sin \pi x\).