Concave Upward and Downward
Concave upward is when the slope increases:  
Concave downward is when the slope decreases: 
What about when the slope stays the same (straight line)? It could be both! See footnote.
Here are some more examples:
Concave Upward is also called Convex, or sometimes Convex Downward
Concave Downward is also called Concave, or sometimes Convex Upward
Finding where ...
Usually our task is to find where a curve is concave upward or concave downward:
Definition
A line drawn between any two points on the curve won't cross over the curve:
Let's make a formula for that!
First, the line: take any two different values a and b (in the interval we are looking at):
Then "slide" between a and b using a value t (which is from 0 to 1):
x = ta + (1−t)b
 When t=0 we get x = 0a+1b = b
 When t=1 we get x = 1a+0b = a
 When t is between 0 and 1 we get values between a and b
Now work out the heights at that xvalue:
When x = ta + (1−t)b:

And (for concave upward) the line should not be below the curve:
For concave downward the line should not be above the curve (≤ becomes ≥):
And those are the actual definitions of concave upward and concave downward.
Remembering
Which way is which? Think:
Concave Upwards = CUP
Calculus
Derivatives can help! The derivative of a function gives the slope.
 When the slope continually increases, the function is concave upward.
 When the slope continually decreases, the function is concave downward.
Taking the second derivative actually tells us if the slope continually increases or decreases.
 When the second derivative is positive, the function is concave upward.
 When the second derivative is negative, the function is concave downward.
Example: the function x^{2}
Its derivative is 2x (see Derivative Rules)
Its second derivative is 2
Both give the correct answer.
Example: f(x) = 5x^{3} + 2x^{2} − 3x
Let's work out the second derivative:
 The derivative is f'(x) = 15x^{2} + 4x − 3 (using Power Rule)
 The second derivative is f''(x) = 30x + 4 (using Power Rule)
And 30x + 4 is negative up to x = −4/30 = −2/15, and positive from there onwards. So:
Note: The point where it changes is called an inflection point.
Footnote: Slope Stays the Same
What about when the slope stays the same (straight line)?
A straight line is acceptable for concave upward or concave downward.
But when we use the special terms strictly concave upward or strictly concave downward then a straight line is not OK.
Example: y = 2x + 1
2x + 1 is a straight line.
It is concave upward.
It is also concave downward.
It is not strictly concave upward.
And it is not strictly concave downward.