Given:
Amplitude = 2
Midline = 5
Period of [tex]\dfrac{3\pi}{4}[/tex].
To find:
The cosine function.
Solution:
The general cosine function is:
[tex]f(x)=A\cos(Bx+C)+D[/tex] ...(i)
Where, |A| is amplitude, [tex]\dfrac{2\pi}{B}[/tex] is period, [tex]-\dfrac{C}{B}[/tex] is phase shift and D is the mid line.
Period of function is [tex]\dfrac{3\pi}{4}[/tex]. So,
[tex]\dfrac{3\pi}{4}=\dfrac{2\pi}{B}[/tex]
[tex]B=2\pi\times \dfrac{4}{3\pi}[/tex]
[tex]B=\dfrac{8}{3}[/tex]
Substituting [tex]A=2,\ B=\dfrac{8}{3},\ C=0,\ D=5[/tex] in (i), we get
[tex]f(x)=2\cos(\dfrac{8}{3}x+0)+5[/tex]
[tex]f(x)=2\cos(\dfrac{8}{3}x)+5[/tex]
Therefore, the required cosine function is [tex]f(x)=2\cos(\dfrac{8}{3}x)+5[/tex].
