Transformation Of Trigonometric Graphs

Transformation to $y = a \sin x$ / $a \cos x$ / $a \tan x$

$\text{amplitude} = \frac{\text{maximum} - \text{minimum}}{2}$

Scaling of graph with a factor of $\frac{1}{b}$ parallel to the $x$-axis

For sin & cos: period becomes $\frac{2\pi}{b}$

For tan: period becomes $\frac{\pi}{b}$

$c = \frac{\text{maximum} + \text{minimum}}{2}$

1. $y = \sin bx$: Scaling of graph with a factor of $\frac{1}{b}$ parallel to the $x$-axis

2. $y = a \sin bx$: Scaling of graph with a factor of $a$ parallel to the $y$-axis (reflecting of graph in $x$-axis if $a < 0$)

3. $y = a \sin bx + c$: Translating of graph by $c$ units parallel to the $y$-axis