Question

What is the parent function of f(x)=3(x-9) to the second power 

Answer 1

$\bf ~~~~~~~~~~~~\textit{function transformations} \\\\\\ f(x)= A( Bx+ C)+ D \\\\ ~~~~y= A( Bx+ C)+ D \\\\ f(x)= A\sqrt{ Bx+ C}+ D \\\\ f(x)= A(\mathbb{R})^{ Bx+ C}+ D \\\\ f(x)= A sin\left( B x+ C \right)+ D\\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf \bullet \textit{ stretches or shrinks horizontally by } A\cdot B\\\\ \bullet \textit{ flips it upside-down if } A\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if } B\textit{ is negative}$

$\bf ~~~~~~\textit{reflection over the y-axis} \\\\ \bullet \textit{ horizontal shift by }\frac{ C}{ B}\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is negative, to the right}\\\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by } D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{ B}$

with that template above in mind, let's see,

$\bf f(x)=\stackrel{A}{3}(\stackrel{B}{1}x\stackrel{C}{-9})+\stackrel{D}{0}$

A = 3, shrunk vertically by 1/3

B = 1, C = -9, C/B = -9, shifted horizontally 9 units to the right.

so f(x) = 3(x-9) is really f(x) = x, in disguise, shrunk and shifted horizontally.