Function behaviors can teach us a lot about their changes by looking at a few important points:

1. Slope of Tangents: The derivative at any point shows us how steep the function is at that spot. If the function is going up, the derivative is positive (we say $f'(x) > 0$). If the function is going down, the derivative is negative ($f'(x) < 0$).

2. Critical Points: These are special spots where the function switches from going up to going down, or the other way around. At these points, the derivative equals zero ($f'(x) = 0$). Finding these critical points helps us discover the highest and lowest points of the function.

3. Concavity: The overall shape of the function tells us something, too. If the second derivative is positive ($f''(x) > 0$), the function opens upwards like a cup. If the second derivative is negative ($f''(x) < 0$), it opens downwards like a cap.

By using these ideas and looking at the graphs, we can better understand how functions work and what their derivatives mean.