What is the equation of the graph below y=cos((x)/(0.4)) y=cos(5x) y=cos(0.4x)

To determine the equation of the graph based on the given details, we need to consider the properties of periodic functions and the cosine function in particular.The cosine function has the general form:\[y = A \cos(Bx - C) + D\]where:- \(A\) is the amplitude (the peak value from the centerline of the wave),- \(B\) is the frequency (which affects the period of the function),- \(C\) is the phase shift (the horizontal shift of the function),- \(D\) is the vertical shift.Given that the function is symmetric about multiple points, including \(x = -3.8\pi, x = -1.8\pi, x = -0.2\pi, x = 0.2\pi, x = 1.8\pi, x = 3.8\pi\), and so on, and that it is monotonically decreasing and increasing in the specified intervals, we can infer that this is a cosine function because the cosine function is symmetric about its maxima and minima.The minimum positive period of the function is given as \(5\pi\). The period of a cosine function is \(\frac{2\pi}{B}\), where \(B\) is the frequency. Therefore, we can set up the equation:\[5\pi = \frac{2\pi}{B}\]Solving for \(B\):\[B = \frac{2\pi}{5\pi} = \frac{2}{5}\]Now, let's examine the given options to see which one matches the frequency we've found:1. \(y = \cos\left(\frac{x}{0.4}\right)\) can be rewritten as \(y = \cos(2.5x)\), which has a frequency of \(2.5\) and a period of \(\frac{2\pi}{2.5} = \frac{4\pi}{5}\), which does not match the given period of \(5\pi\).2. \(y = \cos(5x)\) has a frequency of \(5\) and a period of \(\frac{2\pi}{5}\), which also does not match the given period of \(5\pi\).3. \(y = \cos(0.4x)\) has a frequency of \(0.4\) and a period of \(\frac{2\pi}{0.4} = 5\pi\), which matches the given period.Therefore, the correct equation that matches the given properties of the graph is:\[y = \cos(0.4x)\]Answer: \(y = \cos(0.4x)\)