Graphs and Roots

Every quadratic equation $ax^2 + bx + c = 0$ has a picture: the parabola $y = ax^2 + bx + c$. The roots of the equation are the $x$-intercepts of the graph — the points where $y = 0$.

Example. $y = x^2 - 9$ crosses the $x$-axis where $x^2 - 9 = 0$, i.e. at $x = 3$ and $x = -3$.

Reading the discriminant off the picture:

• two intercepts ↔ $\Delta > 0$
• vertex touching the axis ↔ $\Delta = 0$
• floating clear of the axis ↔ $\Delta < 0$

Symmetry bonus: the axis of symmetry runs exactly midway between the roots at $x = \frac{-b}{2a}$; the vertex (turning point) sits on it. Roots at $\pm 3$ → axis at $x = 0$ → vertex $(0, -9)$.