Question

find a function of the form ( y = a sin ( k x ) + c ) or ( y = a cos ( k x ) + c ) whose graph matches this one:

(leave your answer in exact form; if necessary, type pi for ( pi ).

( y = )

Explanation:

Step1: Find the amplitude \(A\)

The amplitude \(A=\frac{\text{Max}-\text{Min}}{2}\). From the graph, \(\text{Max} = 0\), \(\text{Min}=-4\). So \(A=\frac{0 - (-4)}{2}=2\).

Step2: Find the vertical shift \(C\)

The vertical shift \(C=\frac{\text{Max}+\text{Min}}{2}\). Substituting \(\text{Max} = 0\) and \(\text{Min}=-4\), we get \(C=\frac{0+(-4)}{2}=-2\).

Step3: Find the period \(T\) and \(k\)

The period \(T\) is the distance between two consecutive peaks. From the graph, \(T = 12\). Using the formula \(T=\frac{2\pi}{k}\), we solve for \(k\): \(12=\frac{2\pi}{k}\), so \(k=\frac{\pi}{6}\).

Step4: Determine the function type

Since the graph has a peak at \(x=-13\) and we can use the cosine function (as cosine has a maximum at \(x = 0\) in its basic form \(y=\cos(x)\) and we can adjust the phase - shift. But if we assume no phase - shift (because we can also write it in terms of cosine with the parameters we found), the function is \(y = 2\cos(\frac{\pi}{6}x)-2\).

Answer:

\(y = 2\cos(\frac{\pi}{6}x)-2\)