Question

Why do quadratic equations always have a y-intercept?

Answer 1

Assuming that the parabola is pointing upward or downward, no matter how far away you draw it, since the parabola expands forever, it eventually has to touch the y-axis.

If you think of it algebraically, a parabola is given by the equation y=ax^2+bx+c. If you try to find the y-intercept, you have to substitute 0 for x. When you do, you get y=c. No matter what, there has to be some value for c.