Question

find a function of the form ( y = a sin(kx)+c ) or ( y = a cos(kx)+c ) whose graph matches the function shown below:
leave your answer in exact form; if necessary, type pi for ( pi ).
( y=)
question help: video

Explanation:

Step1: Determine the amplitude \(A\)

The amplitude \(A=\frac{\text{Maximum}-\text{Minimum}}{2}\). From the graph, the maximum value \(y_{max} = 1\) and the minimum value \(y_{min}=-5\). Then \(A=\frac{1 - (-5)}{2}=\frac{6}{2}=3\)

Step2: Determine the vertical shift \(C\)

The vertical shift \(C=\frac{\text{Maximum}+\text{Minimum}}{2}\). So \(C=\frac{1+(-5)}{2}=\frac{-4}{2}=-2\)

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 \(k\). Since \(T = 8\), then \(8=\frac{2\pi}{k}\), and \(k=\frac{\pi}{4}\)

Step4: Determine the function type (sine or cosine)

If we use the sine - function \(y = A\sin(kx)+C\). When \(x = 0\), \(y=-2\). Substitute \(A = 3\), \(k=\frac{\pi}{4}\), \(C=-2\) into \(y = A\sin(kx)+C\), we get \(y = 3\sin(\frac{\pi}{4}x)-2\)

Answer:

\(y = 3\sin(\frac{\pi}{4}x)-2\)