Solution:
Give the following below
[tex]\begin{gathered} amplitude=3 \\ period=\frac{\pi}{2} \\ midline,\text{ }y=-1 \\ Passing\text{ through the point }(0,2) \end{gathered}[/tex]
To find the cosine function, we will apply the general formula for cosine function below
[tex]\begin{gathered} f(x)=a\cos(bx-c)+d \\ Where \\ a\text{ is the amplitude} \\ b\text{ represents the speed of the cycle} \\ d\text{ is the vertical shift} \\ \frac{c}{b}\text{ is the phase shift \lparen horizontal shift\rparen} \\ Period=\frac{2\pi}{b} \end{gathered}[/tex]
To find the value of b
[tex]\begin{gathered} Period=\frac{\pi}{2} \\ \frac{2\pi}{b}=\frac{\pi}{2} \\ Crossmultiply \\ b\times\pi=2\times2\pi \\ b=\frac{4\pi}{\pi} \\ b=4 \end{gathered}[/tex][tex]\begin{gathered} a=3 \\ b=4 \\ d=-1 \\ c=0 \end{gathered}[/tex]
Substitute the values of the variables into the general formula for cosine function
[tex]\begin{gathered} f(x)=a\cos(bx-c)+d \\ f(x)=3\cos(4x-0)-1 \\ f(x)=3\cos4x-1 \end{gathered}[/tex]
Applying a graphing tool,
The graph of one cycle of the function is shown below
