We have a sinusoidal function.
[tex]y=A\cdot\sin (b\cdot x+c)+d[/tex]Its midline is intersected at (0,-3) and has a maximum point at (2,-1.5).
As the midline intersects at y=-3, we know that function has an offset of 3 units down.
This offset is the value of the parameter d, so we have:
[tex]y=A\cdot\sin (b\cdot x+c)-3[/tex]The maximum value happens at point (2,-1.5). The maximum value happens when the pure sin function reaches the value 1, so we can write:
[tex]\begin{gathered} y_{\max }=A\cdot1-3=-1.5 \\ A-3=-1.5 \\ A=-1.5+3 \\ A=1.5 \end{gathered}[/tex]The amplitude is A=1.5, so we can write:
[tex]y=1.5\sin (b\cdot x+c)-3[/tex]We can find the values of the parameters b and c using the x-values of the the points (0,-3) and (2,-1.5):
[tex]\begin{gathered} (x,y)=(0,-3) \\ y(0)=1.5\sin (b\cdot0+c)-3=-3 \\ 1.5\cdot\sin (c)=-3+3 \\ 1.5\cdot\sin (c)=0 \\ \sin (c)=0 \\ c=0 \end{gathered}[/tex][tex]\begin{gathered} (x,y)=(2,-1.5) \\ y(2)=1.5\cdot\sin (b\cdot2)-3=-1.5 \\ 1.5\cdot\sin (2b)=-1.5+3 \\ 1.5\cdot\sin (2b)=1.5 \\ \sin (2b)=1 \\ 2b=\frac{\pi}{2} \\ b=\frac{\pi}{2}\cdot\frac{1}{2} \\ b=\frac{\pi}{4} \end{gathered}[/tex]The function becomes:
[tex]y(x)=1.5\sin (\frac{\pi}{4}x)-3[/tex]The graph of the function is:
Answer: y(x) = 1.5*sin(pi/4 * x)-3
