Sinusoidal Functions

Let the period be $p$ , then we have $f(x+p) = f(x)$ .

$\begin{aligned} f(x+p) & = - 3\cos(\pi(x+{\color{#3D99F6}p})+2) - 6 \\ & = - 3\cos(\pi x+2+{\color{#3D99F6}px}) - 6 \\ & = - 3\cos(\pi x+2) \cos{\color{#3D99F6}px} + 3\sin(\pi x+2) \sin{\color{#3D99F6}px} - 6 & \small \color{#3D99F6} \text{Putting }p=2 \\ & = - 3\cos(\pi x+2) \cos{\color{#3D99F6}2x} + 3\sin(\pi x+2) \sin{\color{#3D99F6}2x} - 6 \\ & = - 3\cos(\pi x+2) - 6 \\ & = f(x) \end{aligned}$

$\implies$ The period $p = \boxed{2}$ .

The general form of a transformed $\sin x$ or $\cos x$ function is

$\displaystyle a\cos[b(x + c)] + d$

where its period equals $\displaystyle \frac{2\pi}{b}$ .

In $\displaystyle f(x) = -3\cos(\pi x + 2) - 6$ ,

$\displaystyle b = \pi$ .

$\displaystyle P = \frac{2\pi}{b} = \frac{2\pi}{\pi} = 2$

The answer is 2.