Is there a family of mathematical curves that looks like the following?

Y1 = f1(X), Y2 = f2(X), …

Y(infinity) = 0, Y(-infinity)=0

-1 <= Y <= 1, or -1 < Y < 1, or combinations thereof

Most importantly,

The first member of the family is roughly bell shaped, like a gaussian density function, and symmetrical and the peak is at (0,1)

The second has a zero at (0,0) and occupies the (-,-) and (+,+) quadrants and is kind of “backwards Z rotated sideways” shaped

The third has two zeros and is kind of M shaped

The nth has n-1 zeros

All members are alternately left-right symmetric or have 180 degree rotational symmetry around the origin

Note, maybe there’s a trivial zeroeth member that is a flat line Y=0 or maybe there’s a reason to say “zeroeth” where I said “first” or something.

I am asking because I think such a family would be very useful for modeling data that are interesting in the neighborhood of 0 but which should have tiny values far from zero. I think for example it’d be nifty to model such data as a linear combination of the first n members of this family.