Transformation Of Trigonometric Graphs

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

1. If $a > 0$: Scaling of graph with a factor of $a$ parallel to the $y$-axis

2. If $a < 0$: Scaling of graph with a factor of $a$ parallel to the $y$-axis, then reflecting of graph in $x$-axis

For sin & cos: amplitude becomes $|a|$

For tan: there is no amplitude

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

🦉 Transformation to $y = \sin bx$ / $\cos bx$ / $\tan bx$

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 $\frac{\pi}{b}$

🦉 Transformation to $y = \sin x + c$ / $\cos x + c$ / $\tan x + c$

Translating of graph by $c$ units parallel to the $y$-axis

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

🦉 Transformation to $y = a \sin bx + c$

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