Hi Chernobog,
Let’s assume (based on your graph) that the domain for this function is -5 <= x <= 5.
Now, the key steps:
1) Recall that if a function’s derivative is negative, then the function itself will be decreasing. In this example, that means we have a function whose graph is always decreasing
2) Also recall that the second derivative tells us about the concavity of the original function—if second derivative is positive, over some interval, then the curve will be concave up (CU) for that interval. Similarly, if the second derivative is negative over some interval, then the curve will be concave down (CD) during that interval.
3) So our requirement boils down to drawing a curve that is always decreasing, and has at least one interval of CU behavior and at least one interval of CD behavior.
4) Our last piece is challenging without a graphing capability. Nonetheless, let’s place the four points on the student’s graph at: A (-5,8), B (-2, 1), C (2,-1) , and D (5, -7).
a. Draw a curve from A to B, that has CU behavior.
b. Draw a curve from B to C, that has CD behavior. So B is a “point of inflection”, that is, a point on the curve where concavity changes, in this case from CU to CD.
c. Remembering to continue the decreasing behavior of our entire curve, connect points C and D with CU behavior. Again, we have a “point of inflection” this time at point C, as the curve changes from CD to CU.
The curve is now complete.
