# How do you find the amplitude and period of y= 2 - 3 cos pi•x?

##### 1 Answer
Jul 7, 2018

The amplitude is $= 5$. The period is $= 2$

#### Explanation:

The amplitude of $\cos x$ is

$- 1 \le \cos x \le 1$

Multiply by $- 3$

$3 \ge - 3 \cos x \ge - 3$

Add #2

$2 + 3 \ge \left(2 - 3 \cos x\right) \ge 2 - 3$

$5 \ge \left(2 - 3 \cos x\right) \ge - 1$

$- 1 \le \left(2 - 3 \cos x\right) \le 5$

The period $T$ of a periodic function $f \left(x\right)$ is

$f \left(x\right) = f \left(x + T\right)$

Therefore,

$\left(2 - 3 \cos \pi x\right) = 2 - 3 \cos \pi \left(x + T\right)$

$\implies$, $- 3 \cos \left(\pi x\right) = - 3 \cos \left(\pi x + \pi T\right)$

$\implies$, $\cos \left(\pi x\right) = \cos \left(\pi x\right) \cos \left(\pi T\right) - \sin \left(\pi x\right) \sin \left(\pi T\right)$

Comparing both sides of the equation

$\left\{\begin{matrix}\cos \pi T = 1 \\ \sin \pi T = 0\end{matrix}\right.$

$\iff$, $\pi T = 2 \pi$

$\implies$, $T = 2$

The period is $= 2$

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