A cosine function is said to have an amplitude of 3, a midline of 5, and a period of 3/4.
It is required to write the cosine function.
Recall that the standard form of a cosine function is:
[tex]y=a\cos(bx+c)+d[/tex]Where a is the amplitude, 2π/b is the period, c is the horizontal shift, and d is the midline or vertical shift.
Equate the given period to 2π/b and solve for b:
[tex]\begin{gathered} \frac{2\pi}{b}=\frac{3}{4} \\ \Rightarrow3b=8\pi \\ \Rightarrow\frac{3b}{3}=\frac{8\pi}{3} \\ \Rightarrow b=\frac{8\pi}{3} \end{gathered}[/tex]Hence, substitute a=3, b=8π/3, and d=5 into the standard form of the cosine function:
[tex]y=3\cos(\frac{8\pi}{3}x+c)+5[/tex]Since it is not given that the cosine function has a horizontal shift, substitute c=0 to get the required function:
[tex]y=3\cos(\frac{8\pi}{3}x)+5[/tex]The function is y=3cos((8π/3) x)+5.
