Example 6 Find the range of the function $y=3 x^{2}-12 x+18 \sqrt{4 x-x^{2}}-23$.


Solve $y=-3\left(\sqrt{4 x-x^{2}}-3\right)^{2}+4$, and the domain of this function is $[0,4], 0 \leqslant \sqrt{4 x-x^{2}} \leqslant$ 2, so, the range of the function $y=3 x^{2}-12 x+18 \sqrt{4 x-x^{2}}-23$ is $[-23,1]$.

Key to solving the problem: Transform the function whose range needs to be determined through substitution or other means into a common function, and use the results related to common functions to obtain the range of the required function. This is an important method for studying the range of functions.