What's the relationship between quadratics and convex functions

Question

Are all quadratic functions convex and vice versa or are quadratic functions just one type of convex function? If so, could someone please provide examples of functions that are convex but not quadratic?

Answer 1

Not all quadratic functions are convex. For instance, $f(x)=-x^2$ is not convex. And not all convex functions are quadratic, like $f(x)=e^x$.

Answer 2

Any quadratic form generalized to $n$ dimensions is of the form $$f(\mathbf x) = \frac{1}{2}\mathbf x^TA\mathbf x + b^T\mathbf x + c$$ This is convex when A is a positive definite matrix.