How are discontinuities and zeros created in rational expressions

I suppose that would just be when the numerator of the rational expression equals zero. Nothing special goes on with the graph in that case; there are simply x-intercepts at those values of x.

Discontinuities are created when the denominator of the rational expression equals zero (because division by zero is undefined). Graphically, this is usually represented by a dashed vertical line indicating a vertical asymptote.