Problem 70 - Precalculus Essentials

Question

Graph one period of each function. $$y=-\left|2 \sin \frac{\pi x}{2}\right|$$

Step-by-Step Solution

Answer

The graph of \(y=-|2 \sin \frac{\pi x}{2}|\) is a wave that oscillates between -2 and 0 for every 4 units on the x-axis. It starts at -2 at x=0, goes up to 0 by x=2, and then goes back down to -2 by x=4.

Step 1: Analyzing the Function

Take a closer look at the function \(y=-|2 \sin \frac{\pi x}{2}|\). The function is a variation of the sine function, with an absolute value applied to it and then made negative. The amplitude (maximum value) of this function is 2 (coming from '2' in '2sin') and the period is \(\frac{2\pi}{(\pi/2)}= 4\) (based on the formula: period \(= \frac{2\pi}{|B|}\), where 'B' is the coefficient of 'x' in the sine function). Since it's the absolute value of the sine function, all negative values will become positive, but due to the negative sign outside the absolute function, the function is reflected over the x-axis.

Step 2: Plotting the sine function

Start by plotting a graph of the original sine function without the modifications, which is \(y = 2 \sin \frac{\pi x}{2}\). The period is 4, meaning by each increment of 4 (0, 4, 8, 12,...) the cycle repeats. Given that the amplitude is 2, the graph will oscillate between -2 and 2.

Step 3: Applying the absolute value function and the reflection

Now apply the absolute value function. This makes all the values of the function positive (or zero). Afterwards, apply the negative sign to the absolute value function. This flips the absolute sine wave over the x-axis.

Step 4: Completing the Graph

The graph now displays a single period of the function \(y=-|2 \sin \frac{\pi x}{2}|\). It shows the oscillation between -2 and 0 over a period of 4 units.