# What is the amplitude of y=cos2x and how does the graph relate to y=cosx?

Oct 5, 2017

For $y = \cos \left(2 x\right)$, $A m p l i t u \mathrm{de} = 1$ & $P e r i o d = \pi$
For $y = \cos x , A m p l i t u \mathrm{de} = 1$ & $P e r i o d = 2 \pi$
Amplitude remains the same but perio halved for $y = \cos \left(2 x\right)$
$y = \cos \left(2 x\right)$
graph{cos(2x) [-10, 10, -5, 5]}
$y = \cos \left(x\right)$
graph{cosx [-10, 10, -5, 5]}

#### Explanation:

$y = a \cdot \cos x \left(b c - c\right) + d$
In given equation $y = \cos \left(2 x\right)$
$a = 1 , b = 2 , c = 0$ & $d = 0$
$\therefore A m p l i t u \mathrm{de} = 1$
$P e r i o d = \frac{2 \pi}{b} = \frac{2 \pi}{2} = \pi$

Similarly for Equation $y = \cos x$,
$A m p l i t u \mathrm{de} = 1$ & $P e r i o d = \frac{2 \pi}{b} = \frac{2 \pi}{1} = 2 \pi$

Period halved to $\pi$ for $y = \cos \left(2 x\right)$ as can be seen from the graph.