A function is concave up if the graph "holds water". The graph would be above its tangent lines.  
A function is concave down if the graph "spills water". The graph would be below its tangent lines.  
An inflection point is the point on the curve where the graph changes from concave up to concave down, or vice versa. 
The sign of the second derivative indicates whether a function is concave up or concave down.
Determine whether the curve below is concave up or concave down at each point given.
y = f(x) = 3x³  6x² at x = 2 [or at the point (2,48)].
Solution:  f '(x) = 9x²  12x f ''(x) = 18x  12 f ''(2) = 36  12 = 48 < 0 The second derivative is negative when x = 2 so the curve is concave down there. 
Use the second derivative as a test function for concave up and concave down. This tells us how to round out a graph when used with the increasing and decreasing information. Points where the second derivative is zero are possible inflection points and places where the graph could change concavity. Continuing the example on Worksheet 41:
Given y = f(x) = 2x³  3x²  72x  4 , find the intervals where f(x) is concave up and where it is concave down.
Steps  Example  
1.  Find f ''(x)  1.  f '(x) = 6x²  6x  72 f ''(x) = 12x  6
 
2.  Find the possible inflection points, where f ''(x) = 0 
2.  f ''(x) = 12x  6 = 6(2x  1) = 0 possible inflection point when x = 1/2  
3.  Put those xvalues on the xaxis and  3.  
4.  Pick one test value in each interval and put it in the 2nd derivative to determine if it is positive or negative. (I'm substituting into the factored form of the 2nd derivative.) 
4. 
 
Answer: 

Note: From the concave up and down information above, we can tell that this function has an inflection point when x = 1/2.
The point of inflection is (1/2,40.5) where the y value is f(1/2).
Combining the above information with the information obtained in Objective 41, we can now sketch a more accurate graph of this function. The only points that must be plotted are the ones corresponding to the critical values. For a more accurate graph, you may also plot the inflection points (or just estimate where they would be) and intercepts. Be sure to put numbers on your axes when you sketch a graph.
Example continued: Sketch y = f(x) = 2x³  3x²
 72x 4
Using those points, sketch a graph which 
xaxis scale = 1 unit yaxis scale = 50 units 
Putting it all together: Sketch the graph of
Find the critical values and test y' inbetween to see if y is increasing or decreasing.
x = 0, x²  3 = 0,
critical values: x = 0, ,
x = 2  x = 1  x = 1  x = 2 
y' = ()  y' = (+)  y' = ()  y' = (+) 
y dec  y inc  y dec  y inc 
Find possible inflection points and test y'' in between to see if y is concave up or down.
x = 1, x = 1, (possible inflection points)
x = 2  x = 0  x = 2 
y'' = (+)  y'' = ()  y'' = (+) 
y c up  y c dwn  y c up 
Plot critical point, then sketch, using information above. For plotting points, use decimal for

From this graph, we can say:
You can sketch a graph even is you only have a description of how it behaves, rather than an equation.
Sketch the graph of a function, f(x), where
Solution:
Without more points, we don't know how steep it rises, but the shape should be about the same. 