Question

how many periods are shown in this graph? -2π 0 2π 4π

Explanation:

Step1: Understand the period of a wave

A period of a periodic function (like a sine or cosine wave) is the length of one complete cycle of the wave. For a standard sine or cosine function, the period is \(2\pi\), but here we can count the number of cycles between the given x - values.

Step2: Analyze the x - axis intervals

The x - axis ranges from \(- 2\pi\) to \(4\pi\). Let's find the total length of the interval: \(4\pi-(-2\pi)=6\pi\).
If we assume the period of the wave (one complete up - and - down cycle) is \(2\pi\) (since from \(-2\pi\) to \(0\) is \(2\pi\), \(0\) to \(2\pi\) is \(2\pi\), and \(2\pi\) to \(4\pi\) is \(2\pi\)).
To find the number of periods, we can also count the number of complete cycles. Looking at the graph:
- From \(-2\pi\) to \(0\): 1 period.
- From \(0\) to \(2\pi\): 1 period.
- From \(2\pi\) to \(4\pi\): 1 period.
Wait, no, let's count the number of peaks and troughs. Each period has one peak and one trough. Let's list the cycles:
First cycle: from \(-2\pi\) to \(0\) (one complete cycle: up, down).
Second cycle: from \(0\) to \(2\pi\) (one complete cycle: up, down).
Third cycle: from \(2\pi\) to \(4\pi\) (one complete cycle: up, down). Wait, no, actually, let's look at the distance between the start and end. The total length is \(4\pi-(-2\pi) = 6\pi\). If the period \(T = 2\pi\), then the number of periods \(n=\frac{6\pi}{2\pi}=3\). Wait, but let's count the number of waves. Looking at the graph, between \(-2\pi\) and \(4\pi\), how many times does the wave repeat?
Looking at the graph:
- From \(-2\pi\) to \(0\): 1 wave (period).
- From \(0\) to \(2\pi\): 1 wave (period).
- From \(2\pi\) to \(4\pi\): 1 wave (period). Wait, no, let's count the number of complete cycles. Let's see the x - values: the first cycle is from \(-2\pi\) to \(0\) (length \(2\pi\)), second from \(0\) to \(2\pi\) (length \(2\pi\)), third from \(2\pi\) to \(4\pi\) (length \(2\pi\)). So the number of periods is \(\frac{4\pi-(-2\pi)}{2\pi}=\frac{6\pi}{2\pi} = 3\). Wait, but maybe I made a mistake. Wait, let's count the number of peaks. Let's see the graph:
- Peak at some point between \(-2\pi\) and \(0\), trough, then peak at some point between \(0\) and \(2\pi\), trough, then peak at some point between \(2\pi\) and \(4\pi\), trough. Wait, no, each period has one peak and one trough. So from \(-2\pi\) to \(0\): peak, trough (1 period). From \(0\) to \(2\pi\): peak, trough (1 period). From \(2\pi\) to \(4\pi\): peak, trough (1 period). So that's 3 periods? Wait, no, wait the distance from \(-2\pi\) to \(4\pi\) is \(6\pi\), and if the period is \(2\pi\), then number of periods is \(6\pi\div2\pi = 3\). But let's check again. Wait, maybe the period is \(2\pi\), so in the interval of length \(6\pi\), the number of periods is \(3\).

Wait, no, let's look at the graph again. The x - axis is marked at \(-2\pi\), \(0\), \(2\pi\), \(4\pi\). So the intervals are \([-2\pi,0]\), \([0,2\pi]\), \([2\pi,4\pi]\). Each interval is length \(2\pi\). Each interval contains one full period (since the wave repeats every \(2\pi\)). So the number of periods is 3? Wait, no, wait the first wave: from \(-2\pi\) to \(0\) (1 period), second from \(0\) to \(2\pi\) (1 period), third from \(2\pi\) to \(4\pi\) (1 period). So total of 3 periods? Wait, but let's count the number of times the wave goes up and down. Let's see the graph:
- From \(-2\pi\): starts, goes up, down, to \(0\): that's 1 period.
- From \(0\): goes up, down, to \(2\pi\): that's 1 period.
- From \(2\pi\): goes up, down, to \(4\pi\): that's 1 period.
So that's 3 periods.

Wait, but maybe I was wrong earlier. Let's calculate the number of periods by the formula \(n=\frac{\text{Total length of the interval}}{\text{Period length}}\). The total length of the interval is \(4\pi-(-2\pi)=6\pi\). If the period of the function (the length of one complete cycle) is \(2\pi\) (because from \(-2\pi\) to \(0\) is \(2\pi\) and that's one cycle, \(0\) to \(2\pi\) is another, \(2\pi\) to \(4\pi\) is another), then the number of periods \(n=\frac{6\pi}{2\pi}=3\).

Answer:

3