[tex]\bf ~~~~~~~~~~~~\textit{function transformations}
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% templates
f(x)= A( Bx+ C)+ D
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~~~~y= A( Bx+ C)+ D
\\\\
f(x)= A\sqrt{ Bx+ C}+ D
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f(x)= A(\mathbb{R})^{ Bx+ C}+ D
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f(x)= A | B x+ C |+ D
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--------------------[/tex]
[tex]\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}\\
~~~~~~\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}\\\\
[/tex]
now, with that template in mind, let's see,
[tex]\bf y=\cfrac{-|x-8|}{2}+1\implies y=-\cfrac{1}{2}|1x-8|+1\\\\\\ y=\stackrel{A}{-\cfrac{1}{2}}|\stackrel{B}{1}x\stackrel{C}{-8}|\stackrel{D}{+1}[/tex]
notice, first off A is negative, reflection over the x-axis.
A is also 1/2, not 1 as in the parent, 1/2 will expand the graph horizontally.
B is unchanged from the parent function as 1.
C is -8, therefore C/B is -8/1 or just -8, namely a horizontal shift to the right by that many units.
D is +1, so a vertical shift upwards of that much.
