Question

what is the period? ? π

Explanation:

Step1: Recall the definition of period

The period of a periodic function is the horizontal length over which the function repeats its pattern. For a trigonometric function (or a periodic graph), we can find the period by looking at the distance between two consecutive peaks (or troughs) or by analyzing the x - axis intervals.

Step2: Analyze the graph's x - axis intervals

Looking at the x - axis labels: from \(-\frac{3\pi}{2}\) to \(-\pi\), \(-\pi\) to \(-\frac{\pi}{2}\), \(-\frac{\pi}{2}\) to \(0\), \(0\) to \(\frac{\pi}{2}\), \(\frac{\pi}{2}\) to \(\pi\), \(\pi\) to \(\frac{3\pi}{2}\). Let's count the number of periods in a given interval. Let's take the interval from \(0\) to \(\pi\). How many periods are there in the interval from \(0\) to \(\pi\)? Looking at the graph, we can see that between \(0\) and \(\pi\), there are 2 full periods.

The formula for the period \(T\) of a function with a repeating pattern over an interval \(\Delta x\) with \(n\) periods is \(T=\frac{\Delta x}{n}\). Here, \(\Delta x=\pi - 0=\pi\) and \(n = 2\). So the period \(T=\frac{\pi}{2}\). We can also check by looking at the distance between two consecutive peaks. If we take two adjacent peaks, the horizontal distance between them is \(\frac{\pi}{2}\).

Answer:

The period is \(\frac{\pi}{2}\), so the numerator is \(1\) and the denominator is \(2\). So the answer is \(\frac{1\pi}{2}\), so the box for the numerator is \(1\) and the box for the denominator is \(2\). (If we consider the form \(\frac{?\pi}{[\ ]}\), then \(? = 1\) and \([\ ]=2\))