When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?

Question

When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?

Why can't the outside factor of $3$ simply be used for the dilation factor from the $x$-axis?

Answer 1

Think about what happens to $x$ in the expression. First you multiply $x$ by $2$, then subtract $3$, then square the result, multiply by $3$, and finally subtract $\frac{1}{4}$. There have been two multiplications in this process, each one dilating the $x$-axis. To take $3$ as the dilation factor would ignore the fact that $x$ was multiplied by $2$ at the very beginning.