2009
10.29

The graph of the quadratic function $\emph{y} = \emph{f}(x) = ax^{2} + bx + c$ is parabola.

• If $a > 0$, the parabola opens upwards. If $a < 0$, it opens downwards
• The vertex is $\left ( -\frac{b}{2a} , \: \emph{f}\left ( -\frac{b}{2a} \right ) \right )$
• The y-interacept = $c$
• The x-intercepts are obtained by setting $\emph{y} = 0$ and solving for $x$

This relationship between the value inside the square root (the discriminant), the type of solutions (two different solutions, one repeated solution, or no real solutions), and the number of x-intercepts (on the corresponding graph) of the quadratic is summarized in this table:

 $x^{2} - 2x - 3$ $x^{2} - 6x + 9$ $x^{2} + 3x + 3$ a positive number inside the square root zero inside the square root a negative number inside the square root two real solutions one (repeated) real solution two complex solutions two distinct x-intercepts one (repeated) x-intercept no x-intercepts

References: