The discriminant, shown as $D = b^2 - 4ac$, is very important when we graph quadratic functions.

Here’s how it helps us understand the roots of the equation:

• Roots of the Equation:

  • If $D > 0$: There are two different real roots. This means the graph will cross the x-axis at two points.
  • If $D = 0$: There is one real root. This is known as a double root because the graph touches the x-axis at just one point.
  • If $D < 0$: There are no real roots. This means the graph (or parabola) does not cross the x-axis at all.

• Finding the Vertex: We can find where the vertex (the highest or lowest point on the graph) is located by using the formula $x = -\frac{b}{2a}$. The values of $b$ and $a$ from $D$ help determine this position.

By understanding the discriminant, we can graph parabolas more accurately and figure out where they intersect with the x-axis.