The following curve turns once. It changes sign twice and the sign of its gradient changes once, giving rise to four distinct regions, as shown in the image below:

The graph of a quadratic that dips below the x axis at the bottom of it. Its sign and the sign of its gradient in different regions are shown.

What are the different possibilities of sign combination for a curve which turns exactly once?

How can you be sure that you have found them all?

  • Which other curves turn just once?

  • How can you sketch a curve where the sign of the gradient changes at some point?

  • What happens to your results if you shift the axes up or down?