merging the equations of the line

Question

I have two equations for plotting the line. How can I merge them to get a single equation? $$\begin{align}y&=7.515x^3 - 10.229x^2 +5.05x, \,\, x \leq 0.5\\ y&=0.185x + 0.815, \,\, 0.5<x\leq1\end{align}$$

I actually require equation of a line passing through following points: $(0,0),\,(0.2,0.7), \,(0.5,0.9),\,(1,1)$

But connecting lines between these points should be straight, especially between the last two points.

Answer 1

If you just want to find a polynomial curve that passing through these 4 points, this example could be helpful. The fitting curve can be calculated by Lagrange Interpolation. In your case, I find the curve below by using Mathematica's function InterpolatingPolynomial that could satisfy your conditions.

$$y = (x-1) ((5.08333 (x-0.5)-1.6) x+1)+1 = 5.08333 x^3-9.225 x^2+5.14167 x$$