Answer:
[tex]g(x)=4sin(2x+\frac{\pi}{2})+3[/tex]
Step-by-step explanation:
Because the function intersects its midline at [tex](\frac{3\pi}{4},3)[/tex], then the midline is [tex]d=3[/tex].
Additionally, the amplitude is just the positive distance between the maximum/minimum and the midline, so the amplitude is [tex]a=7-3=4[/tex].
Also, given that period is [tex]\frac{2\pi}{b}[/tex] and the fact that the period is [tex]\pi-0=\pi[/tex] from our given maximum, we have the equation [tex]\frac{2\pi}{b}=\pi[/tex] where [tex]b=2[/tex].
Lastly, to find [tex]c[/tex], we know that the phase shift, [tex]-\frac{c}{b}[/tex], is [tex]-\frac{\pi}{4}[/tex] (or [tex]\frac{\pi}{4}[/tex] to the left) since [tex]\pi-\frac{3\pi}{4}=-\frac{\pi}{4}[/tex]. Therefore, we have the equation [tex]-\frac{c}{2}=-\frac{\pi}{4}[/tex] where [tex]c=\frac{\pi}{2}[/tex].
Putting it all together, our final equation is [tex]g(x)=4sin(2x+\frac{\pi}{2})+3[/tex]
