= Solution
Assign one unit parameter interval to each piece. Then
$$
\boxed{S(u)=\begin{cases}F(u),&0\le u\le1,\\E(u-1),&1\le u\le2.\end{cases}}
$$
There is no ambiguity at $u=1$, since both pieces give the origin. For a single <B-spline> representation take degree two, the four <de Boor points>
$$
(D_0,D_1,D_2,D_3)=((-4,-1),(0,1),(0,-1),(4,1))
$$
and the equally spaced <spline knot> vector $\tau=(-2,-1,0,1,2,3,4)$. With $N_{i,2}$ denoting the normalized degree-two <B-spline> on this vector, the complete valid parameter interval is $[\tau_2,\tau_4]=[0,2]$ and
$$
S(u)=\sum_{i=0}^3D_iN_{i,2}(u).
$$
On $[0,1]$ the active handles are $D_0,D_1,D_2$, and on $[1,2]$ they are $D_1,D_2,D_3$. Shared handles and a simple interior <spline knot> are what merge the two pieces into one uniform <B-spline>. The exterior <spline knots> support the basis; they do not extend the interval being drawn.
In particular, the requested horizontal pieces and their vertical counterparts are
$$
\boxed{S_x(u)=\begin{cases}-2(1-u)^2,&0\le u\le1,\\2(u-1)^2,&1\le u\le2,\end{cases}\qquad
S_y(u)=\begin{cases}2u(1-u),&0\le u\le1,\\2(u-1)(u-2),&1\le u\le2.\end{cases}}
$$
\Image[/past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/geometric-modelling-2015-spline.png]
{title=Two quadratic pieces, their Bézier handles, the shared uniform B-spline control polygon, and the vertical tangent at the join}
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