Problem:
Explore the graph of the function

f(x) = x4 - 3 x2

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

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 appears to be perpendicular to and is bisected by the y-axis. This implies that the points (a, f(a)) and (-a, f(-a)) are symmetric with respect to the y-axis.

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 y-axis.

You can change the function in the notebook above. Try

• x4- 9x2
[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.]
• x6- 9x2
• x4- 9x3
• x6- 9x4
[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.]