  We show how to obtain linear combinations of polynomials in an orthogonal
sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k
a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal
polynomials of order $k\le n-1$. The polynomials in the sequence
$\{Q_{n,k}\}_{n\geq 0}$ are obtained from $P_{n}$, by making use of parameter
shifts. We use an algorithmic approach to find these linear combinations for
each family applicable and these equations are used to prove
quasi-orthogonality of order $k$. We also determine the location of the extreme
zeros of the quasi-orthogonal polynomials with respect to the end points of the
interval of orthogonality of the sequence $\{P_n\}_{n\geq 0}$, where possible.
