 # Determine the equation of the function with the following properties: A Lawson Cain 2022-02-13 Answered
Determine the equation of the function with the following properties:
A transformation of the basic $y=\mathrm{cot}\left(x\right)$ function with
- A period of $\frac{\pi }{3}$
- A stretching factor of 3
- A phase shift to the left by straight $\frac{\pi }{12}$
- A vertical shift down by 5
Write the equation for the function described above:
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$y=\mathrm{cot}\left(x\right)$
$t=a\mathrm{cot}\left(b\left(x-h\right)\right)+c$
$a=\text{streching factor}=3$
$b=\text{period}$
$\text{Period}=\frac{\pi }{b}=\frac{\pi }{3}$
$b=3$
$h=\text{phase shift(left side)}=\frac{-\pi }{12}$
$c=\text{vertical shift down}=-5$
$y=a\mathrm{cot}\left(b\left(x-h\right)\right)+c$
$=3\mathrm{cot}\left(3\left(x+\frac{\pi }{12}\right)\right)-5$
$y=3\mathrm{cot}\left(3x+\frac{\pi }{4}\right)-5$
$y=3\mathrm{cot}\left(3x+\frac{\pi }{4}\right)-5$