Respuesta :
[tex]\bf ~~~~~~~~~~~~\textit{function transformations}
\\\\\\
f(x)=Asin(Bx+C)+D
\\\\
f(x)=Acos(Bx+C)+D\\\\
f(x)=Atan(Bx+C)+D
\\\\
-------------------\\\\
\bullet \textit{ stretches or shrinks}\\
~~~~~~\textit{horizontally by amplitude } |A|\cdot B\\\\
\bullet \textit{ flips it upside-down if }A\textit{ is negative}\\
~~~~~~\textit{reflection over the x-axis}
\\\\
\bullet \textit{ flips it sideways if }B\textit{ is negative}\\
~~~~~~\textit{reflection over the y-axis}[/tex]
[tex]\bf \bullet \textit{ horizontal shift by }\frac{C}{B}\\ ~~~~~~if\ \frac{C}{B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{C}{B}\textit{ is positive, to the left}\\\\ \bullet \textit{vertical shift by }D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}\\ ~~~~~~\frac{2\pi }{B}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ ~~~~~~\frac{\pi }{B}\ for\ tan(\theta),\ cot(\theta)[/tex]
with that template in mind, let's check this one,
[tex]\bf y=tan(\stackrel{B}{2}x\stackrel{C}{-\pi })\\\\ -------------------------------\\\\ \textit{horizontal shift of }\cfrac{C}{B}\implies \cfrac{-\pi }{2},\textit{ so of }\frac{\pi }{2}\textit{ to the right} \\\\\\ \textit{horizontal shrinkage of }B\implies 2, \textit{so by }\frac{1}{2}[/tex]
[tex]\bf \bullet \textit{ horizontal shift by }\frac{C}{B}\\ ~~~~~~if\ \frac{C}{B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{C}{B}\textit{ is positive, to the left}\\\\ \bullet \textit{vertical shift by }D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}\\ ~~~~~~\frac{2\pi }{B}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ ~~~~~~\frac{\pi }{B}\ for\ tan(\theta),\ cot(\theta)[/tex]
with that template in mind, let's check this one,
[tex]\bf y=tan(\stackrel{B}{2}x\stackrel{C}{-\pi })\\\\ -------------------------------\\\\ \textit{horizontal shift of }\cfrac{C}{B}\implies \cfrac{-\pi }{2},\textit{ so of }\frac{\pi }{2}\textit{ to the right} \\\\\\ \textit{horizontal shrinkage of }B\implies 2, \textit{so by }\frac{1}{2}[/tex]
