Problem:
Explore the graph of the function

f(x) = x3 - 3 x

for values of x between -2 and 2 to view geometrically that the graph of this function is symmetric with respect to the origin.

Visualization:
In the following LiveMath Notebook, you can view the graph of y = f(x) (blue graph) and a straight line segment (red graph) between the points (a, f(a)) (green point) and (-a, f(-a)) (yellow point). Note that this line segment contains the origin which appears to be the midpoint of the line segment. This implies that the points (a, f(a)) and (-a, f(-a)) are symmetric with respect to the origin.

Note that as you change the value of a, the points (a, f(a)) and (-a, f(-a)) still appear to be symmetric with respect to the origin.

You can change the function in the notebook above. Try

• x3- 9x
[You will need to zoom out to see the entire line segment. To zoom out, click on the 'rocket ship' on the upper left-hand corner of the graph.]
• x5- 9x
• x3- 9x2
• x5- 9x3
[To change a coefficent in the definition of f(x) in the notebook above, highlight the coefficent with the mouse and then type in the new coefficient. To change x to x2, highlight x in the notebook with the mouse and press the shift and 6 keys at the same time followed by 2.]