Stretching and shifting trig graphs

Two words first:

The amplitude is how far the wave goes above (or below) its centre line. $y = \sin x$ goes from $-1$ to $1$ , so its amplitude is 1.
The period is the width of one full wave before it repeats. For $y = \sin x$ it's $360^\circ$ .

Try it with numbers.

$y = 3\sin x$ : every height is tripled. It now goes from $-3$ to $3$ . Amplitude 3, period still $360^\circ$ .
$y = \sin 2x$ : the angle runs twice as fast, so one wave takes half the width. Period $360^\circ \div 2 = 180^\circ$ , amplitude still 1.

Every change to $y = \sin x$ is a transformation you already know from the transformations lessons. Here they all are ( $\lvert a \rvert$ means the size of $a$ ignoring its sign):

Graph	Transformation	Effect
$y = a\sin x$	stretch parallel to the $y$ -axis, factor $a$	amplitude $\lvert a \rvert$ , range $-\lvert a \rvert$ to $\lvert a \rvert$
$y = \sin bx$	stretch parallel to the $x$ -axis, factor $\tfrac1b$	period $\tfrac{360^\circ}{b}$ (or $\tfrac{2\pi}{b}$ )
$y = \sin x + c$	translation $c$ up	centre line $y = c$
$y = \sin(x + d)$	translation $d$ to the left	everything slides sideways
$y = -\sin x$	reflection in the $x$ -axis	maxima become minima

The same works for cos, and for tan (which has no amplitude, but stretches and shifts just the same).

Same moves as topic 1.2, trig flavour