Question

Write a cosine function that has an amplitude of 2, a midline of 5 and a period of \frac{3\pi}{4} 4 3π ​ .

Answer 1

Given:

Amplitude = 2

Midline = 5

Period of $\dfrac{3\pi}{4}$.

To find:

The cosine function.

Solution:

The general cosine function is:

$f(x)=A\cos(Bx+C)+D$            ...(i)

Where, |A| is amplitude, $\dfrac{2\pi}{B}$ is period, $-\dfrac{C}{B}$ is phase shift and D is the mid line.

Period of function is $\dfrac{3\pi}{4}$. So,

$\dfrac{3\pi}{4}=\dfrac{2\pi}{B}$

$B=2\pi\times \dfrac{4}{3\pi}$

$B=\dfrac{8}{3}$

Substituting $A=2,\ B=\dfrac{8}{3},\ C=0,\ D=5$ in (i), we get

$f(x)=2\cos(\dfrac{8}{3}x+0)+5$

$f(x)=2\cos(\dfrac{8}{3}x)+5$

Therefore, the required cosine function is $f(x)=2\cos(\dfrac{8}{3}x)+5$.