A polynomial graph meets the $x$-axis at exactly 2 points but is NOT a parabola (it bends more than once).
What is the smallest degree such a graph can have?

A parabola (degree 2) bends only once. A curve that bends more than once needs degree at least 3, and a cubic can cross the $x$-axis at exactly 2 points (one crossing plus one touch), so smallest degree is 3.