SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the general form of a sinusoidal function
[tex]\begin{gathered} f(x)=Asin(Bx+c)-d \\ \text{where A is the amplitude} \\ Period=\frac{2\pi}{B} \\ c\text{ is the phase shift} \\ \text{d is the vertical shift} \end{gathered}[/tex]
STEP 2: Write the given values
[tex]\begin{gathered} midline=(0,-6) \\ minimum=(2.5,-9) \end{gathered}[/tex]
STEP 3: find the amplitude
[tex]\begin{gathered} A=absolute\text{ }difference\text{ between the y-values of the minimum point and the midline} \\ A=|-9-(-6)|=|-9+6|=|-3|=3 \end{gathered}[/tex]
STEP 4: Get the value of B
[tex]\begin{gathered} Period=4\times2.5=10 \\ B=\frac{2\pi}{10}=\frac{\pi}{5} \end{gathered}[/tex]
STEP 5: Find the value of c
The function was shifted by 5 units to the right, hence, the value of c is -5
STEP 6: Find the value of the Vertical shift
It can be seen from the midline that the function was shifted by 6 units downwards, therefore, d =6.
STEP 7: Get the sinusoidal function by joining these terms
Hence, the sinusoidal function is given by:
[tex]f(x)=3\sin(\frac{\pi}{5}(x-5))-6[/tex][tex]f(x)=3\sin(\frac{\pi}{5}(x-5))-6[/tex]
[tex]f(x)=3\sin(\frac{\pi}{5}(x-5))-6[/tex]
