Problem 14 - Precalculus Essentials

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 is 2 and the period is 2. The function forms a 'reflective' sine curve over one period from x=0 to x=2.

Step 1: Determine the Amplitude

The amplitude is given by the absolute value of the coefficient of the \(\sin\) function. Here, our coefficient is -2, hence the amplitude is \(|-2|\) which equals to 2.

Step 2: Calculate the Period

The period of a sine function is given by the formula \( \frac{2 \pi}{\text{coefficient of } x} \). In our function, the coefficient of \(x\) is \(\pi\), so the period would be \( \frac{2\pi}{\pi} = 2 \).

Step 3: Graphing the Function

To graph one period of this function: (a) Draw an x-y plane. (b) Mark points along the x-axis with a distance equal to the period. As our period is 2, we could mark points at 0, 0.5, 1, 1.5 and 2. (c) The function forms a sine wave starting from the origin (0,0). As the coefficient of \(\sin\) is negative, we expect a 'reflection' of the normal \(\sin\) function. Thus start at (0,0), it dips down to the minima at (0.5,-2), comes back to (1,0), goes to the maxima at (1.5,2) and returns to (2,0), completing one period of the function.