Proof to the properties of sinusoidal functions A sinusoidal function is defined if $f(x)$ can be written as $a\sin [b(x+c)]+d$ where $a$ is the amplitude of the curve, $b$ is related to the period, $c$ is the horizontal shift and $d$ relents the vertical shift of the curve.
But can anyone explain me how $a,b,c,d$ represents those?
 A: $$a\sin [b(x+c)]+d$$
a - the magnitude of vertical oscillation. Effect is stretch/shrinking vertically
b - the frequency, or how long is its cycle. Effect is the stretching/shrinking horizontally
c - the horizontal shift of the curve 
d - the vertical shift of the curve
EDIT
For better understanding of b
Let's first take a step back, and consider the linear function $f(x)=bx $, and it is obvious this function is a continuous linear transformation of $x$, and the effective is to stretch $x$. For example, if $x \in (0,\pi)$, and $b=2$, then $bx \in (0,2\pi)$
Now let's check $\sin(bx)= \sin \circ f (x)$
$\sin x$ is periodic, and $\sin (x) = \sin (x + 2\pi)$
Now due the the stretching effect of $f(x)=bx $, we notice that whenever $x$ changes by amount of $2\pi/b$, $f(x) $ changes by amount of $2\pi$ (let's assume $b\ne 0$ here)
Put above formally into equations 
$$\sin(b\cdot (x + \frac{2\pi}{b}))=\sin(bx + b \cdot \frac{2\pi}{b})=\sin(bx + 2\pi)= \sin(bx)$$
Thus now, the periodicity becomes $\frac{2\pi}{b}$
