Zeros, x-Intercepts, and the y-Intercept

Solving f(x) = 0 and finding where a curve crosses the x-axis are the same question asked two ways. The x-axis is where every point has height zero, so any x that makes f(x) equal zero names a spot on that axis the curve actually touches. Each solution becomes a point (x, 0) on the graph. The y-intercept is a different animal: it's the one point where the input x is 0, giving the point (0, f(0)). Same function, two separate questions: what input gives output zero, versus what output comes from input zero.

Take f(x) = 2x² - 8x - 10. Factor to get 2(x - 5)(x + 1), so the zeros are x = 5 and x = -1, meaning the graph crosses the x-axis at (5, 0) and (-1, 0). The y-intercept comes from plugging in x = 0: f(0) = -10, giving (0, -10). For f(x) = x³ - 4x, factor as x(x - 2)(x + 2), so the curve crosses the x-axis at x = 0, 2, and -2. A repeated factor like (x - p)² makes the curve touch the axis and bounce back instead of crossing, because the sign of f(x) doesn't flip there; an odd power like (x - p) or (x - p)³ lets it pass straight through.

This is why factored form is worth hunting for. Once a quadratic or cubic is written as a product of linear factors, each factor hands you a zero directly, and the shape near each zero (cross or bounce) depends on whether its exponent is odd or even. That link between algebraic factors and graphical behavior is the same idea that carries forward into sketching polynomials and reasoning about their end behavior.