When looking at the graphs of quadratic functions, the x-intercepts are important!

The x-intercepts are where the graph crosses the x-axis. They give us helpful information about the solutions to the quadratic equation.

Understanding Roots and X-Intercepts

1. Roots of a Quadratic: The roots of a quadratic equation, like $y = ax^2 + bx + c$, are the values of $x$ that make $y$ equal to 0.

   In simpler terms, these are the solutions to the equation $ax^2 + bx + c = 0$.

2. X-Intercepts: You can find the x-intercepts by solving for $x$ when $y$ is 0.

   This brings us to the quadratic formula:

   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

   The two answers you get from this formula are the x-coordinates of the x-intercepts on the graph.

What Do These Intercepts Reveal?

• Number of Roots:

  • If there are two x-intercepts, the quadratic has two different real roots. For example, in the equation $y = x^2 - 5x + 6$, the x-intercepts are at $x = 2$ and $x = 3$.

  • If there is one x-intercept, the quadratic has one real root (which is a repeated root). This means the graph just touches the x-axis at that point. For instance, in the equation $y = x^2 - 4x + 4$, there’s one x-intercept at $x = 2$.

  • If there are no x-intercepts, the roots are complex. This means the parabola doesn't touch the x-axis at all. An example of this is $y = x^2 + 1$, which has no real solutions.

Visualizing the Concept

When you look at a graph:

• Parabolas that open upwards (when $a > 0$) can have two, one, or zero x-intercepts.
• You can clearly see each situation, helping you understand how the vertex's position and the direction of the parabola relate to the roots.

Conclusion

So, the x-intercepts show where a quadratic function crosses the x-axis.

They also tell us about how many roots the function has.

Knowing this can really help you understand and draw the graphs of quadratic functions better!