Concave up and concave down? 
I am unable to find out how to find concave up and concave down just by looking at a graph of function $f(x)$. I was told that if the tangent line of the slope can be visualized under the graph, it's concave up, and if it's visualized above, it's concave down, but it doesn't appear to be so. I put concave up as $(0,3)$ and concave down as $(3,6)$. How would you identify concave up and down? 
 A: Here is one way to visualize concave up and concave down.
Imagine that your graph is on a map, with the positive $y$ direction pointing north and the positive $x$ direction pointing east. Think of an ant car (a tiny car driven by an ant) driving from west to the east along the graph of your curve.
The graph is concave up if the steering wheel of the car is to the left of center--in other words, if the car is turning to its left. The graph is concave down if the steering wheel is to the right of center--in other word, if the car is turning to its right.
In your graph, the ant car starts at $x=0$ and moves generally to the right (east). You see at first that the car is turning slightly to its left, then as $x$ gets to $1$ the car is turning even more to the left. As it passes $x=1$ the car straightens out, and is driving straight ahead at $x=2$. After that, the car is steering a little to its right and is turning much to its right at $x=3$. And so on.
So we see that the function $f(x)$ is concave up on the interval $[0,2)$.
Can you continue from there? There are multiple other ways to think of concavity, but I find that my students understand this pretty easily.
A: Suppose you pour water into a cylinder of such cross section, 
ConcaveUp trickles water down the trough and holds water in the tub.
ConcaveDown  trickles water away and spills out, water falling  down.
In the first case slope is <0 to start with, increases to 0 and next becomes > 0.
In the second  case slope is >0 at start, decreases to 0 and next becomes < 0.
