Step 1: Write out the general sine function
[tex]y=A\sin (B(x+C))+D[/tex][tex]\begin{gathered} \text{ Where} \\ A=\text{ the amplitude of the sine function} \\ \frac{2\pi}{B}=\text{ the period of the function} \\ C=\text{ the phase shift} \\ D=\text{ the vertical shift} \end{gathered}[/tex]Step 2: Write out the values from the graph and substitute them into the general sine function
In this case,
[tex]A=1[/tex][tex]\begin{gathered} \text{ the period is }\frac{\pi}{2} \\ \text{ therefore } \\ \frac{2\pi}{B}=\frac{\pi}{2} \\ \pi B=4\pi \\ \text{ thus} \\ B=4 \end{gathered}[/tex]The period is π / 2. Therefore,
[tex]\begin{gathered} BC=\frac{\pi}{2} \\ \text{ thus} \\ C=\frac{\pi}{2B}=\frac{\pi}{2\times4}=\frac{\pi}{8} \end{gathered}[/tex]The vertical shift is downward by 2 units. Therefore,
[tex]D=-2[/tex]Hence, the equation of the trigonometric graph is given by
[tex]\begin{gathered} y=\sin (4(x+\frac{\pi}{8}))-2 \\ \text{ thus} \\ y=\sin (4x+\frac{\pi}{2})-2 \end{gathered}[/tex]y = sin (4x + p