Investigation on

by

Ana Kuzle

I begin my investigation by plugging in several values for a.

So, we can see that the value of a influences whether the graph of the function is above or under x-axis. If a>0, then the graph is above x-axis because a is positive and x^2 is positive for all real x. Therefore, value of y is always positive.

If a<0, then the graph is under x-axis because a is negative and x^2 is positive for all real x. Therefore, value of y is always negative.

Animation shows it very nice.

The graph of a given function has only one zero-point and that point (0, 0) is because .

I continue my investigation on how a>0 influences given function.

We can see also in the animation that the value of a influences how 'open' is the graph. As positive a becomes bigger the graph is less open, i.e. it gets closer to y-axis. As positive a becomes smaller, the graph is more open, i.e. it gets more close to x-axis (grows more rapidly to infinity).

The same reasoning applies for a<0. But, as negative a becomes bigger the graph is more open, i.e. it gets closer to x-axis (grows more rapidly to negative infinity). As positive a becomes smaller, the graph is less open, i.e. it gets less close to y-axis as shown here.

From both cases we can see that the graphs are symmetrical with y-axis being its axis of symmetry. Why is that?

. Thus, is an even function.

Afterwards, I was interested how opposite values of a influence graph of .

We can see that for given opposite values of a, graphs of the given function are symmetrical where x-axis is its axis of symmetry.