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
$\text{amplitude}=\dfrac{\text{maximum} -\text{minimum}}{2}$
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
$\text{amplitude}=\dfrac{\text{maximum} -\text{minimum}}{2}$
Transformation to $y= sin bx / cos bx / tan bx$
Scaling of graph with a factor of $\dfrac{1}{b}$ parallel to the $x$-axis
For sin & cos: period becomes $\dfrac{2\pi}{b}$
For tan: period becomes $\dfrac{\pi}{b}$
For sin & cos: period becomes $\dfrac{2\pi}{b}$
For tan: period becomes $\dfrac{\pi}{b}$
Transformation to $y=sin x +c/ cos x+c / tan x+c$
Translating of graph by $c$ units parallel to the $y$-axis
$c=\dfrac{\text{maximum}+\text{minimum}}{2}$
$c=\dfrac{\text{maximum}+\text{minimum}}{2}$
Transformation to $y=a sin bx+c$
1. $y=sin bx$: Scaling of graph with a factor of $\dfrac{1}{b}$ 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
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
