Question

170 points possible answered: 5/17
question 6
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 = )
add work

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Explanation:

Step1: Determine the amplitude \(A\)

The amplitude \(A=\frac{\text{Max}-\text{Min}}{2}\). From the graph, \(\text{Max} = 3\) and \(\text{Min}=-1\). So \(A=\frac{3 - (-1)}{2}=\frac{4}{2}=2\).

Step2: Determine the vertical shift \(C\)

The vertical shift \(C=\frac{\text{Max}+\text{Min}}{2}\). Substituting \(\text{Max} = 3\) and \(\text{Min}=-1\), we get \(C=\frac{3+(-1)}{2}=\frac{2}{2}=1\).

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

The period \(T\) is the distance between two consecutive peaks. From the graph, \(T = 8\). Using the formula \(T=\frac{2\pi}{k}\), we solve for \(k\): \(8=\frac{2\pi}{k}\), so \(k=\frac{2\pi}{8}=\frac{\pi}{4}\).

Step4: Choose the function form

Since the graph has a maximum at \(x = 0\), the cosine function is more appropriate. The general form \(y = A\cos(kx)+C\) gives \(y = 2\cos(\frac{\pi}{4}x)+1\).

Answer:

\(y = 2\cos(\frac{\pi}{4}x)+1\)