Linear Noether normalization for a hypersurface

For a nonconstant polynomial $f\in k[t_1,\dots ,t_n]$ over an infinite field, an invertible linear change of coordinates can make $f$ monic in the final coordinate after multiplication by a scalar. Therefore $k[t_1,\dots ,t_n]/(f)$ is integral over a polynomial algebra in the first $n-1$ new coordinates.