CMOQR 2017 Problem 4

In this question we re-define the operations addition and multiplication as follows: $a + b$ is defined as the minimum of $a$ and $b$ , while $a * b$ is defined to be the sum of $a$ and $b$ . For example, $3+4 = 3$ , $3*4 = 7$ , and $$3*4^2+5*4+7 = \min(\text{3 plus 4 plus 4}, \text{5 plus 4}, 7) = \min(11, 9, 7) = 7.$$ Let $a, b, c$ be real numbers. Characterize, in terms of $a, b, c$ , what the graph of $y = ax^2+bx+c$ looks like.