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34kurt
2 years ago
7

Describe the graph of y={1/(2x-10)}-3 compared to the graph of y=1/x

Mathematics
1 answer:
tester [92]2 years ago
7 0
\bf ~~~~~~~~~~~~\textit{function transformations}
\\\\\\
% templates
f(x)=  A(  Bx+  C)+  D
\\\\
~~~~y=  A(  Bx+  C)+  D
\\\\
f(x)=  A\sqrt{  Bx+  C}+  D
\\\\
f(x)=  A(\mathbb{R})^{  Bx+  C}+  D
\\\\
f(x)=  A sin\left( B x+  C  \right)+  D
\\\\
--------------------

\bf \bullet \textit{ stretches or shrinks horizontally by  }   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}

\bf ~~~~~~\textit{reflection over the y-axis}
\\\\
\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{ period of }\frac{2\pi }{  B}

with that template in mind, let's check these two

\bf \stackrel{parent}{y=\cfrac{1}{x}}\qquad \qquad\qquad \qquad  \stackrel{transformed}{y=\cfrac{1}{\stackrel{B}{2}x\stackrel{C}{-10}}\stackrel{D}{-3}}\\\\
-------------------------------\\\\
B=2\qquad \textit{shrinks horizontally by }\frac{1}{2}
\\\\\\
C=-10\qquad \cfrac{C}{B}=\cfrac{-10}{2}\implies -5\qquad \textit{horizontally right-shifted by }5
\\\\\\
D=-3\qquad \textit{vertically down-shifted by }3
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