Problem 12 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=3 \sin 2 \pi x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=3 \sin 2 \pi x\) is 3, and the period is 1. The graph of one period of this function will show one full wave oscillating between -3 and 3, over an interval of length 1 on the x-axis.

Step 1: Determine the amplitude

The amplitude (A) of a sine function is the absolute value of the coefficient of \(\sin\). For the provided function \(y=3 \sin 2 \pi x\), the coefficient of \(\sin\) is 3, hence the amplitude A = 3.

Step 2: Determine the period

The period (T) of a sine function is given by \(T=2\pi /B\), where B is the coefficient of x. In our function, that value is \(2\pi\), so period T = \(2\pi /2\pi = 1\).

Step 3: Graph the function

To graph one period of the function, first draw a horizontal line (the x-axis). Then, mark a point at the height of the amplitude (3) above the x-axis. This will be the maximum value. Also mark a point at the same distance below the x-axis, this will be the minimum value. The function will oscillate between these two values. Then, mark points along the x-axis at intervals of the period (1). In one cycle, the sine function starts from zero, goes up to the maximum value, comes back to zero, goes down to the minimum value, and returns to zero. Connect these points with a smooth curve to complete one period of the sine function.