Question

find the period of the sinusoidal function.
simplify any fractions.
period =

Explanation:

Step1: Identify key points

A sinusoidal function's period is the horizontal distance between two consecutive peaks (or troughs). Let's look at the graph. From the grid, we can see the distance between two consecutive peaks (or troughs) needs to be calculated.

Step2: Analyze the grid

Looking at the x - axis, the total length from \(- 2\pi\) to \(2\pi\) is \(4\pi\). Now, let's count the number of periods in this interval. From the graph, we can see that in the interval from \(-2\pi\) to \(2\pi\) (length \(4\pi\)), how many full periods are there? Let's see the pattern. If we look at the troughs or peaks, we can see that the distance between two consecutive peaks (or troughs) is \(\frac{2\pi}{3}\)? Wait, no, wait. Wait, let's re - examine. Wait, the x - axis has markings at \(-2\pi,-\pi,0,\pi,2\pi\). Let's look at the number of cycles between \(-2\pi\) and \(2\pi\). Let's count the number of periods. From the graph, we can see that between \(-2\pi\) and \(2\pi\) (a length of \(4\pi\)), how many full cycles? Wait, maybe a better way: the period is the length of one full cycle. Let's find two consecutive identical points, like two consecutive troughs. Let's take the trough at \(x = - 2\pi\) and the next trough. Wait, no, let's look at the grid. Each square: let's assume the distance between \(-2\pi\) and \(2\pi\) is divided into, say, 12 units (since from \(-2\pi\) to \(2\pi\) is \(4\pi\), and if we count the number of periods, let's see. Wait, maybe the graph has a period of \(\frac{2\pi}{3}\)? No, wait, let's do it properly. The formula for the period of a sinusoidal function \(y = A\sin(Bx + C)+D\) or \(y = A\cos(Bx + C)+D\) is \(T=\frac{2\pi}{|B|}\). But from the graph, let's find the distance between two consecutive peaks. Let's look at the x - coordinates of two consecutive peaks. Suppose one peak is at \(x = - \frac{4\pi}{3}\) and the next at \(x = - \frac{4\pi}{3}+\frac{2\pi}{3}=-\frac{2\pi}{3}\), then the next at \(x = 0\)? No, that doesn't seem right. Wait, maybe I made a mistake. Wait, let's look at the number of periods in the interval from \(-2\pi\) to \(2\pi\). Let's count the number of cycles. From the graph, we can see that in the interval \([-2\pi,2\pi]\), there are 6 cycles? Wait, no, let's count the number of peaks. From \(-2\pi\) to \(2\pi\), how many peaks? Let's see, the graph has peaks, and if we count the number of peaks between \(-2\pi\) and \(2\pi\), let's say there are 6 peaks. Then the length of the interval is \(4\pi\), so the period \(T=\frac{4\pi}{6}=\frac{2\pi}{3}\)? No, that's not right. Wait, wait, maybe I messed up. Wait, the x - axis: from \(-2\pi\) to \(2\pi\) is \(4\pi\) units. Let's look at the graph again. Let's take two consecutive troughs. The trough at \(x=-2\pi\), then the next trough at \(x=-2\pi + \frac{2\pi}{3}\)? No, that's not. Wait, maybe the correct way is: let's find the distance between two consecutive identical points. Let's take the point where the graph crosses a certain level, like the mid - line. Wait, the mid - line of the sinusoidal function: the maximum value is - 4? Wait, no, the graph is below the x - axis? Wait, the y - axis has values from - 6 to 6. Wait, the troughs are at \(y=-6\) and peaks at \(y = - 4\)? Wait, maybe I misread the graph. Wait, the graph is a sinusoidal curve with troughs at \(y=-6\) and peaks at \(y = - 4\). Let's find the distance between two consecutive troughs. Let's look at the x - coordinates. Let's take the trough at \(x=-2\pi\) and the next trough. Let's see the grid: from \(-2\pi\) to \(2\pi\) is \(4\pi\). Let's count the number of periods in this interval. If we look at the graph, we can see that there are 6 periods in the interval from \(-2\pi\) to \(2\pi\) (since \(4\pi\) length with 6 cycles). Then the period \(T=\frac{4\pi}{6}=\frac{2\pi}{3}\)? No, that's not. Wait, wait, maybe the number of periods is 3? No, let's do it step by step.

Wait, another approach: the period is the horizontal distance between two consecutive peaks (or two consecutive troughs). Let's look at the graph. Let's take a peak (the highest point of a cycle) and the next peak. From the grid, we can see that the distance between two consecutive peaks is \(\frac{2\pi}{3}\)? No, wait, let's count the number of units between \(-2\pi\) and \(2\pi\). The total length is \(2\pi-(-2\pi)=4\pi\). Now, looking at the graph, how many full cycles are there in this \(4\pi\) length? Let's count the number of times the graph repeats. Let's see, from the left - most part to the right - most part, we can see that there are 6 full cycles. So the length of one cycle (period) is \(\frac{4\pi}{6}=\frac{2\pi}{3}\)? No, that can't be. Wait, maybe I made a mistake in counting the number of cycles. Wait, let's look at the x - axis labels: \(-2\pi,-\pi,0,\pi,2\pi\). Let's look at the graph between \(-\pi\) and \(\pi\). How many cycles? From \(-\pi\) to \(\pi\) is \(2\pi\) length. If we count the cycles in \(2\pi\) length, how many? Let's see, the graph has a period of \(\frac{2\pi}{3}\), so in \(2\pi\) length, the number of cycles is \(\frac{2\pi}{\frac{2\pi}{3}} = 3\) cycles. Wait, but maybe the correct period is \(\frac{2\pi}{3}\)? No, wait, let's check with the standard. Wait, maybe the graph is \(y=-5 - \cos(3x)\) or something, so the period would be \(\frac{2\pi}{3}\). Wait, let's confirm. The period of \(y = \cos(Bx)\) is \(\frac{2\pi}{|B|}\). If \(B = 3\), then period is \(\frac{2\pi}{3}\). Looking at the graph, if we have three periods in the interval from \(-\pi\) to \(\pi\) (length \(2\pi\)), then \(3\times T=2\pi\), so \(T = \frac{2\pi}{3}\). Yes, that makes sense. So the period is \(\frac{2\pi}{3}\)? Wait, no, wait, maybe I messed up. Wait, let's take two consecutive troughs. Let's say the first trough is at \(x=-2\pi\), the next at \(x=-2\pi+\frac{2\pi}{3}\), then at \(x=-2\pi + \frac{4\pi}{3}\), then at \(x=-2\pi + 2\pi=-2\pi + \frac{6\pi}{3}=\frac{2\pi}{3}\)? No, that's not right. Wait, maybe the correct period is \(\frac{2\pi}{3}\). Wait, let's look at the grid again. Each square: let's assume that the distance between \(-2\pi\) and \(2\pi\) is divided into 12 small squares (since from \(-2\pi\) to \(2\pi\) is \(4\pi\), and if each small square is \(\frac{\pi}{3}\) in length). Then the distance between two consecutive peaks is 2 small squares? No, \(\frac{\pi}{3}\times2=\frac{2\pi}{3}\). Yes, that seems right. So the period of the sinusoidal function is \(\frac{2\pi}{3}\)? Wait, no, wait, maybe I made a mistake. Wait, let's do it again. The period is the length of one full cycle. Let's find two points where the function has the same value and slope (e.g., two consecutive peaks). The x - coordinate of one peak and the x - coordinate of the next peak. From the graph, we can see that the distance between two consecutive peaks is \(\frac{2\pi}{3}\). So the period \(T=\frac{2\pi}{3}\). Wait, but let's check with the interval from \(-2\pi\) to \(2\pi\). The length is \(4\pi\). If the period is \(\frac{2\pi}{3}\), then the number of periods in \(4\pi\) is \(\frac{4\pi}{\frac{2\pi}{3}}=6\), which matches the number of cycles we can see in the graph. So the period is \(\frac{2\pi}{3}\).

Answer:

\(\frac{2\pi}{3}\)