In a first calculus course, it is not unusual for students to encounter the theorems which state: If f is an even (odd) differentiable function, then its derivative is odd (even). In our paper, we prove some theorems which show how the symmetry of a continuous function f with respect to (i) the vertical line: x = a or (ii) with respect to the point: (a, 0), determines the symmetry of the antiderivative of f defined by F(x) = integral [superscript x] [subscript a] of f(t)dt + F(a). We conclude with an example that shows how our results lead to a "two-line proof" that the graph of any cubic function is symmetric with respect to its point of inflection.