Question

what is the period of the sinusoidal function? enter your answer in the box.

Explanation:

Step1: Identify key points on the graph

Looking at the sinusoidal graph, we can observe the distance between two consecutive peaks or troughs. Let's take two consecutive troughs (or peaks) to calculate the period. For example, one trough is at \( x = -8 \) and the next trough is at \( x = -4 \), then the next at \( x = 0 \), then \( x = 4 \), etc.

Step2: Calculate the period

The period is the horizontal distance between two consecutive identical points (like troughs or peaks). The distance between \( x=-8 \) and \( x = -4 \) is \(|-4 - (-8)|= 4\)? Wait, no, wait. Wait, looking at the graph, let's check the distance between two consecutive cycles. Wait, maybe I made a mistake. Wait, let's look at the x - axis. The graph repeats every how many units? Let's see the points: from \( x=-8 \) to \( x = -4 \) is 4? No, wait, let's check the distance between two consecutive peaks or troughs. Wait, another way: the period of a sinusoidal function \( y = A\sin(Bx + C)+D \) or \( y = A\cos(Bx + C)+D \) is given by \( T=\frac{2\pi}{|B|}\), but from the graph, we can calculate it by looking at the horizontal length of one full cycle. Let's look at the graph: from \( x = -8 \) to \( x = -4 \), is that a full cycle? Wait, no. Wait, let's take two points where the graph repeats. Let's take the point at \( x=-8 \) (a trough) and the next trough at \( x = -4 \)? No, that's not right. Wait, wait, looking at the graph, the distance between \( x=-8 \) and \( x = -4 \) is 4? Wait, no, let's check the grid. The x - axis has marks at -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10. Let's take two consecutive peaks or troughs. Let's take the trough at \( x=-8 \) and the next trough at \( x=-4 \)? No, that's 4 units? Wait, no, that can't be. Wait, maybe I misread. Wait, let's take the peak at \( x=-7 \) (no, the graph is a sine - like curve). Wait, another approach: the period is the length of one full cycle. Let's look at the graph from \( x = -8 \) to \( x = -4 \): no, that's not a full cycle. Wait, wait, the graph at \( x=-8 \) is a trough, at \( x=-4 \) is a trough? No, looking at the graph, the troughs are at \( x=-8, x = -4, x = 0, x = 4, x = 8\). So the distance between \( x=-8 \) and \( x=-4 \) is \(|-4-(-8)| = 4\)? Wait, no, \( -4-(-8)=4\), so the distance between two consecutive troughs is 4 units? Wait, but let's check the peak. The peaks are at \( x=-7, x=-3, x = 1, x = 5, x = 9\). The distance between \( x=-7 \) and \( x=-3 \) is \(|-3-(-7)| = 4\). So the period is 4? Wait, no, wait, that seems too small. Wait, maybe I made a mistake. Wait, let's look at the graph again. Wait, the x - axis: from -10 to 10, with grid lines at every 2 units? Wait, no, the labels are at -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10. So the distance between -8 and -4 is 4 units. But let's check the number of cycles between -8 and 0. From -8 to 0 is 8 units, and how many cycles? From -8 to -4 is one cycle? No, that can't be. Wait, no, wait, the period of a sinusoidal function is the distance between two consecutive identical points (like two consecutive peaks or two consecutive troughs). Let's take the trough at \( x=-8 \) and the next trough at \( x=-4 \): the difference is \( -4-(-8)=4 \). Wait, but let's check with another pair. The trough at \( x=-4 \) and the next at \( x = 0 \): \( 0 - (-4)=4 \). The trough at \( x = 0 \) and the next at \( x = 4 \): \( 4-0 = 4 \). So the period is 4? Wait, but that seems short. Wait, maybe the grid is such that each square is 2 units? No, the labels are at -10, -8, -6, etc., so the distance between -8 and -6 is 2 units. Wait, let's count the number of units between two consecutive troughs. The trough at \( x=-8 \) and the trough at \( x=-4 \): the difference is \( -4-(-8)=4 \), but if each grid line is 2 units, then from -8 to -4 is 2 grid spaces (since -8 to -6 is 1, -6 to -4 is 2). Wait, no, the x - axis labels are at -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10. So the distance between -8 and -4 is 4 units (since \( -4-(-8)=4 \)). So the period of the sinusoidal function is 4? Wait, no, wait, maybe I made a mistake. Wait, let's look at the graph again. The graph has a cycle from, say, \( x=-8 \) to \( x=-4 \), then \( x=-4 \) to \( x = 0 \), etc. So the length of one full cycle (period) is 4? Wait, but let's check the standard sine function. The standard sine function \( y=\sin(x) \) has a period of \( 2\pi\approx6.28 \), but this is a horizontally compressed or stretched sine function. From the graph, the distance between two consecutive troughs (or peaks) is 4 units. So the period is 4.

Answer:

4