Let $\Pi$ be a partition of $[a,b]$ with points $x_0,\dots,x_n$. If $s\in C^1[a,b]$ and on every subinterval $[x_{i-1},x_i]$ $s$ is a quadratic spline with the following interpolation knots, $$s(z_i)=y_i,\quad (z_i)=\frac{(x_{i-1}+x_i)}{2} \quad i=1,\dots,n$$ and $$s(x_0)=y_0 \quad s(x_n)=y_{n+1}.$$ Give a way of calculating the the splines $s_i$.
I tried using $s(x)=ax^2+bx+c$ and plugging in $$s(x_i)=ax_i^2+bx_i+c,~s(x_{i-1})=ax_{x-i}^2+bx_{i-1}+c$$ and similarly for $z_i$ $$s(z_i)=a\frac{(x_{i-1}-x_i)^2}{4}+b\frac{(x_{i-1}+x_i)}{2}+c$$ not sure where to go from here.
You need a different $a,b,c$ for each spline segment. Call these $a_i$, $b_i$, $c_i$. Then, the equations you wrote down form a linear system. You can solve to get the $a_i$, $b_i$, $c_i$. The system is banded, so it's easy to solve.