Question

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Explanation:

Step1: Identify the function type

The graph is a periodic wave, symmetric about the y - axis, with amplitude 1, period \(2\pi\), and minimum value - 1, maximum value 1. The general form of a cosine function is \(y = A\cos(Bx)+C\). Here, \(A = 1\), \(B = 1\), \(C = 0\), so the function is \(y=\cos(2x)\)? Wait, no, let's check the period. The distance between two consecutive peaks (e.g., from \(-\pi\) to \(\pi\)) is \(2\pi\)? Wait, no, from \(-\pi\) to \(\pi\) is \(2\pi\), but the period of \(y = \cos(x)\) is \(2\pi\), but here the graph has a peak at \(-\pi\), \(\pi\) and trough at \(0\), \(2\pi\), \(- 2\pi\). Wait, actually, the function \(y=\cos(2x)\) has period \(\pi\), but here the period between \(-2\pi\) and \(0\) and \(0\) and \(2\pi\) - no, looking at the x - axis labels: the distance between \(-2\pi\) and \(0\) is \(2\pi\), but the graph completes two cycles between \(-2\pi\) and \(2\pi\). Wait, the period \(T\) is the distance between two consecutive peaks. The peak at \(-\pi\) and the next peak at \(\pi\), so \(T=\pi - (-\pi)=2\pi\)? No, \(\pi-(-\pi) = 2\pi\), but the number of cycles between \(-2\pi\) and \(2\pi\) is 2. Wait, maybe it's \(y = \cos(x)\) shifted? No, the standard cosine function \(y = \cos(x)\) has a maximum at \(x = 0\), but here at \(x = 0\) it's a minimum. Oh! So it's \(y=-\cos(x)\)? Wait, no, at \(x = 0\), \(y=- 1\), at \(x=\pi\), \(y = 1\), at \(x = -\pi\), \(y = 1\). So the function is \(y = -\cos(x)\)? Wait, no, let's check the values: when \(x = 0\), \(y=-1\); \(x=\pi\), \(y = 1\); \(x = -\pi\), \(y = 1\); \(x = 2\pi\), \(y=-1\); \(x=-2\pi\), \(y = - 1\). So the function is \(y=-\cos(x)\)? Wait, no, the standard cosine function \(y=\cos(x)\) has \(y(0) = 1\), \(y(\pi)=-1\). So if we have \(y = -\cos(x)\), then \(y(0)=-1\), \(y(\pi)=1\), which matches the graph. Alternatively, it could be \(y=\cos(x+\pi)\) (using the phase shift formula \(y = \cos(x + \phi)\), \(\phi=\pi\) gives \(y=-\cos(x)\)).

Step2: Determine the function equation

The general form of a cosine function is \(y = A\cos(Bx - C)+D\).

• Amplitude \(A\): The maximum value is \(1\), minimum is \(-1\), so \(A=\frac{1 - (-1)}{2}=1\).
• Period \(T\): The distance between two consecutive peaks (e.g., \(x = -\pi\) and \(x=\pi\)) is \(T = \pi-(-\pi)=2\pi\). The formula for period is \(T=\frac{2\pi}{|B|}\), so \(2\pi=\frac{2\pi}{|B|}\), which gives \(|B| = 1\), so \(B = 1\) (we can take \(B = 1\) for simplicity).
• Vertical shift \(D\): The midline is \(y = 0\) (since \(\frac{1+(-1)}{2}=0\)), so \(D = 0\).
• Phase shift \(C\): The standard cosine function \(y=\cos(x)\) has a maximum at \(x = 0\), but our graph has a minimum at \(x = 0\). So we can write it as \(y=-\cos(x)\) (which is equivalent to \(y=\cos(x+\pi)\), so \(C=-\pi\) or \(C=\pi\) depending on the formula).

So the function represented by the graph is \(y = -\cos(x)\) or \(y=\cos(x+\pi)\).

Answer:

The function of the graph is \(y = -\cos(x)\) (or equivalent forms like \(y=\cos(x + \pi)\))