Problem 11 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=4 \sin \pi x$$

Step-by-Step Solution

Answer

The amplitude is 4 and the period is 2. The graph will show a sine wave with a peak at y=4, a trough at y=-4, completing one wave cycle in an interval of 2 units along x-axis.

Step 1: Find the Amplitude

The amplitude is determined by the absolute value of the coefficient of the sine function. In the case of the function \( y = 4 \sin \pi x \), the amplitude is \( |4| \), which equals 4.

Step 2: Find the Period

The period of a sine or cosine function is given by \( \frac{2\pi}{|B|} \), where B is the coefficient of x inside the function. For the function \( y = 4 \sin \pi x \), the period is \( \frac{2\pi}{|\pi|} = 2 \).

Step 3: Graph the Function

With amplitude of 4 and period of 2, graph the function. The function starts at 0, reaches a maximum value of +4, goes back to 0, reaches a minimum value of -4, and returns to 0, in the interval of 2 units along x-axis. This interval of 2 completes one period of the sine function.