Convexify $x\le a+by^2$

Question

I have the following non-convex constraint: $$ x\le a+by^2\quad\text{where}\quad a,b>0,\,y\in[0,y_{max}]\text{ and }a\approx by_{max}^2 $$ On a drawing, it looks something like this:

The above figure also shows a possible convexification, but one that I don't like because for small $y$ it allows some infeasible $x$ values and for large $y$ it forbids having some feasible $x$ values.

My question: can I do better? Thanks for helping!

Answer 1

A convex relaxation of the constraint would give you the convex hull of the feasible region: in this case $$z \le a + b y y_{max}$$