Answer:
The required function is [tex]y=\sin\left(\frac{\pi}{7}x\right)-3[/tex].
Step-by-step explanation:
From the given graph it is clear that the value of function is not extreme at x=0, so the required function is a sine function.
The general form of a sine function is
[tex]y=A\sin(kx)+C[/tex] ..... (1)
where, A is amplitude, [tex]\frac{2\pi}{k}[/tex] is period and C is midline.
From the given graph it is clear that the maximum value of the function is -2 and minimum value of the function is -4.
[tex]Amplitude=\frac{Maximum-Minimum}{2}[/tex]
[tex]Amplitude=\frac{-2-(-4)}{2}=1[/tex]
[tex]Midline=\frac{Maximum+Minimum}{2}[/tex]
[tex]Midline=\frac{-2+(-4)}{2}=-3[/tex]
The function complete a cycle in 14 units, so period of the function is 14.
[tex]\frac{2\pi}{k}=14[/tex]
[tex]\frac{2\pi}{14}=k[/tex]
[tex]\frac{\pi}{7}=k[/tex]
Substitute A=1, [tex]k=\frac{\pi}{7}[/tex] and C=-3 in equation (1).
[tex]y=(1)\sin(\frac{\pi}{7}x)+(-3)[/tex]
[tex]y=\sin\left(\frac{\pi}{7}x\right)-3[/tex]
Therefore the required function is [tex]y=\sin\left(\frac{\pi}{7}x\right)-3[/tex].
