Problem 70 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y=-\left|2 \sin \frac{\pi x}{2}\right|\) is an inverted sine curve starting from \(x=0\) to \(x=4\) on the x-axis, oscillating downwards due to the absolute value and the negative value.

Step 1: Understand the function

The student is dealing with a sine function which is multiplied by 2, has an absolute value, and its outcome is furthermore negated. The \(\frac{\pi x}{2}\) part inside the sine function also affects the period of the sine wave. The negative sign at the front of the absolute value function flips it downwards.

Step 2: Identify the period of the function

The period depends upon the coefficient of \(x\) inside the sine function. Normally, the period of \(\sin(x)\) is \(2\pi\). If we have a coefficient \(b\) multiplied with \(x\), the period becomes \(\frac{2\pi }{ b}\). Here, the coefficient of \(x\) is \(\frac{\pi}{2}\), so the period of the function is \(\frac{2\pi }{ \frac{\pi}{2}} = 4\). So, the graph of the function repeats after every 4 units.

Step 3: Calculate key points for one period and plot

We can calculate key points in all significant places in a period (start - mid - end points as well as tips). Starting from \(x=0\) to \(x=4\), calculate \(y\) values for \(x = 0, 1, 2, 3, 4\) and plot these points. Note that the absolute value of \(y\) will always be positive, but due to the negative sign, the graph will lie below the x-axis. Thus, our graph will be an inverted wave under the x-axis from \(x = 0\) to \(x = 4\).

Step 4: Draw the graph of the function

Finally, draw the graph using these key points. Make sure you just draw one period of the function, from \(x=0\) to \(x=4\). The graph should oscillate below the x-axis, reflecting the fact that the function is the absolute value of a trigonometric function and it's negative.