Q:

Constructing piecewise quadratic polynomial

The question asks to construct a piecewise quadratic polynomial defined on the interval $\mathbb{R}$ of the form
$$
B_0=
\begin{cases}
p(x)=x^2,\qquad\qquad\quad\; 0\leq x<1,\\
q(x)=ax^2+bx+c,\quad\, 1\leq x<2,\\
r(x)=(3-x)^2,\qquad\quad 2\leq x<3,\\
0\qquad\qquad\qquad\qquad\quad\,\text{otherwise}
\end{cases}
$$
I need to find the coefficients for $q(x)$. However, I'm having trouble with figuring out which method to use. We're dealing with uniform quadratic B-splines. Would I need to use the Gram-Schmidt method?
Thanks. 

A:

From my comments above, you require the spline to satisfy $p(1)=q(1)$, $q(2)=r(2)$ for function continuity and $p'(1)=q'(1)$ and $q'(2)=r'(2)$ for derivative continuity.
The result of this is an overdetermined linear system, which has a unique solution.

