One of the best ways for students to understand concavity in graphs is by using the second derivative test. Let’s break it down:

1. What is Concavity?:

   • When the second derivative, written as $f''(x)$, is positive (this means $f''(x) > 0$), the graph is concave up. You can picture this like a bowl that is turned right-side up.
   • When the second derivative is negative (when $f''(x) < 0$), the graph is concave down. Think of this as a bowl that is flipped upside down.

2. Examples with Graphs:

   • For the function $f(x) = x^3$, its second derivative is $f''(x) = 6x$. This changes at $x=0$. On the left side of zero, the graph is concave down. On the right side, it’s concave up.
   • If we look at $f(x) = -x^2$, the second derivative is $f''(x) = -2$. Since this value is always less than zero for any $x$, the graph is always concave down.

By looking at these changes and shapes in the graphs, students will find it much easier to understand the idea of concavity!