the x- and y-intercepts

The intercepts of a curve are the places where it crosses the axes — the simplest landmarks for sketching any graph by hand. The x-intercepts are where the curve meets the x-axis, and the y-intercepts are where it meets the y-axis; spotting them gives you two or three anchor points before you draw a single other thing.

The defining feature of the x-axis is that every point on it has y = 0; likewise every point on the y-axis has x = 0. So the method is simply: to find x-intercepts, set y = 0 in the equation and solve for x; to find y-intercepts, set x = 0 and solve for y. For the line 2x + 3y = 12: setting y = 0 gives 2x = 12, x = 6, so the x-intercept is (6, 0); setting x = 0 gives 3y = 12, y = 4, so the y-intercept is (0, 4). Two points are enough to draw a whole line. For a curve there may be several x-intercepts (where a parabola dips through the axis) or none at all.

Intercepts carry meaning beyond sketching. In y = m x + b the number b is the y-intercept, the starting height; the x-intercepts of a function are its zeros or roots, the inputs that make the output zero — the single most asked-for feature in applied problems ('when does the profit hit zero?', 'where does the projectile land?'). One honest note: a curve need not have a y-intercept if it never crosses x = 0 (a vertical line x = 3 does not), and it need not have any x-intercept (the circle x^2 + y^2 = 1 shifted far up may miss the x-axis entirely).