Question

(coWrite a cosine function that has an amplitude of 2, a midline of 5 and a period of 8.

Answer 1

Given

A cosine function has an amplitude of 2, a midline of 5 and a period of 8.

To find:

A cosine function.

Explanation:

Let the function be,

$y=A\cos(wx+\varphi)$

Since amplitude is 2.

Then,

$A=2$

Also, since the period is 8.

Then,

$\begin{gathered} w=\frac{2\pi}{8} \\ =\frac{\pi}{4} \end{gathered}$

And, since midline is 5.

Then,

$\begin{gathered} \pi=\frac{\pi}{4}x+\varphi \\ \varphi=\pi-\frac{\pi}{4}x \\ Since\text{ }midline\text{ }is\text{ }5. \\ Then,\text{ }x=5 \\ \Rightarrow\varphi=\pi-\frac{5\pi}{4} \\ \Rightarrow\varphi=\frac{4\pi-5\pi}{4} \\ \Rightarrow\varphi=-\frac{\pi}{4} \end{gathered}$

Hence, the cosine function is,

$y=2\cos(\frac{\pi}{4}x-\frac{\pi}{4})$