Find the domain and range of $LaTeX: \displaystyle f(x)=- 4 x^{2} + 8 x + 9$

The domain of every polynomial is all real numbers. The sign of $LaTeX: \displaystyle a$ is negative so the function has a maximum value at the vertex. The vertex is located at $LaTeX: \displaystyle h=\frac{-b}{2a}=1$ and $LaTeX: \displaystyle k=f(h)=13$ . This gives the range as $LaTeX: \displaystyle (-\infty,13]$