Aitor’s theorem

Sometimes magic happens in class: good moments of reasoning appear in classroom instantly, without any planification. Sometimes one student asks you for a reasoning validation and then you are marvelled. This happened to me two weeks ago. We were revising homework: sketching graph of quadratic functions with their concavity, their vertex and the cutting points to axes… then, Aitor said:

But it’s needless. We don’t need to calculate the cutting points to $x$ -axis because the vertex is “positive”

What?, I said. Can you explain it with more detail?

And then, we put in blackboard what we have known as Aitor’s theorem. We wrote in blackboard its hypothesis and its thesis:

Hypothesis: The parabola is concave and the $y$ of the vertex is positive
Thesis: the parabola has not cutting points of $x$ -axis

General view of the board. You can see zoomed version

Zoom at Aitor’s theorem text. You can see zoomed version

Since then, we have applied this theorem a lot of times for saving us the time to calculate the cutting points of $x$ -axis and, more important, my students know firsthand why mathematical reasoning is.