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.