Chapter 4 - Vector Spaces - 4.2 Vector Spaces - 4.2 Exercises - Page 160: 20

It is not a vector space.

The set of all quadratic functions whose graphs pass through the origin with the standard operations is not a vector space because it is not closed under addition. For example, $f(x)=x^2+x$ and $g(x)=-x^2+2x$ are two quadratic functions passing through the origin but $f(x)+g(x)=3x$ which is not quadratic.