Question

b. $per = \underline{\quad\quad} a = \underline{\quad\quad}$ equation (as $\cos x$):$\underline{\quad\quad}$

Explanation:

Step1: Find the period

The period of a cosine function \(y = A\cos(Bx)\) is \(Per=\frac{2\pi}{|B|}\). From the graph, the distance between two consecutive peaks (or troughs) is \(6.28319 - (- 6.28319)=12.56638\) (but wait, no. Looking at the standard cosine \(y = \cos x\) has period \(2\pi\approx6.28319\). Here, the graph shows one - cycle from \(-6.28319\) to \(6.28319\), but actually, if we consider the general form. Wait, no, looking at the x - axis, the function repeats every \(6.28319-(-6.28319) = 12.56638\)? No, wait, no. Wait, the standard period of \(y=\cos x\) is \(2\pi\approx6.28319\). But if we look at the graph, from \(x = - 6.28319\) to \(x=6.28319\) is one full cycle. So \(Per = 12.56638\approx4\pi\). Wait, no, wait, the formula for the period of \(y = A\cos(Bx)\) is \(T=\frac{2\pi}{|B|}\). If we assume the general form \(y = A\cos(Bx)+k\). But looking at the x - values: the function from \(x=-6.28319\) to \(x = 6.28319\) is a full cycle. So \(Per=12.56638 = 4\pi\) (since \(2\pi\approx6.28319\)).

Step2: Find the amplitude

The amplitude \(A\) of a cosine function \(y = A\cos(Bx)+k\) is given by \(A=\frac{\text{Max}-\text{Min}}{2}\). The maximum value of the function \(y = 2\) and the minimum value \(y=-2\). So \(A=\frac{2 - (-2)}{2}=2\)

Step3: Find the equation

The general form of a cosine function is \(y = A\cos(Bx)+k\). We know \(A = 2\), \(k = 0\) (since the mid - line \(y=\frac{2+( - 2)}{2}=0\)), and \(Per=\frac{2\pi}{|B|}=4\pi\), so \(|B|=\frac{2\pi}{4\pi}=\frac{1}{2}\). The equation is \(y = 2\cos(\frac{1}{2}x)\)

Answer:

\(Per = 4\pi\), \(A = 2\), Equation: \(y = 2\cos(\frac{1}{2}x)\)