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 ?