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The problem I have is the following:

Given three $x,y$ points, $(-1, -y_{0})$, $(0,0)$ and $(1,y_{2})$, where $y_{0} geq 0$ and $y_2 geq 0$, I want to interpolate smoothly between them with a function or spline that is monotonic, has continuous first and second derivatives, and with the following conditions imposed on the derivatives at the end points: $f'(-1) = y_{0}$, $f'(+1) = y_{2}$, $f''(-1) = f''(1) = 0$. In other words I have two lines $y=mx$, with in general different gradients, one defined for $x < 1$ and the other for $x > 1$, which need to be smoothly joined in the intermediate region, like in this sketch.

I suspect this can be solved with splines of a sufficiently high order, but so far I've not been able find an example in the literature which clearly covers the case with all of the monotonicity, continuous 1st/2nd derivatives and the end-point conditions imposed.

Grateful for a solution or even a pointer in the right direction - I don't come from a very mathematical background so a more explicit solution would certainly be appreciated.