Let us look at the graph first. This is a typical graph which is expected for korean high school students to draw with pencil and paper. Though Algebra Xpressor draws the graph easily it is really hard for students to make a conjecture about the shape of the graph. Here the idea of derivative is very useful. The given function is , and its differentiated function is . Let us graph the differentiated function with Algebra Xpressor. Two graphs are drawn together for help of understanding. The derivative function says that will have a symmetric figure. So once students can draw on part of symmetric figure, then the other part will be easily drawn. When 0<x<1.5, the graph will have positive slope lines, and as x increases the graph is going to be parallel to x-axis since the slopes of tangent lines is getting close to 0. When x=0, then y=0 , and when x=1, then y=1 by .
Now, I am interested in the second derivative which will esplain about the first derivative function. Next graph is the second differentiated function . Do you think that you can give meaningful information from the second differentiated function to students?
For further investigation related with similar types of fuctions I introduce a function . Even though students are not allowed to graph the function they may guess the shape of it. Because the format of function is very similiar with . It just has a difference in one of coefficients. Let's make a conjecture, then try to draw the graph. The graph is following. Now, it's a problem for you to solve. Which one is and which one is ?