In dealing with quadratic equations, we often see them written like this:

$$ax^2 + bx + c = 0$$

Here, the letters $a$, $b$, and $c$ are important because they help us find the solutions (or roots) of the equation and how the graph of the function looks.

1. The Role of $a$ (Leading Coefficient)

• Direction of the Parabola:
  • If $a$ is greater than 0 (like $a = 2$), the parabola opens up.
  • If $a$ is less than 0 (like $a = -2$), it opens down.

This changes where the roots, or x-intercepts, will be.

• Width of the Parabola:
  • The size of $a$ also affects how wide or narrow the parabola appears.
  • A bigger absolute value of $a$ (for example, $a = 3$ instead of $a = 1$) makes the parabola narrower, bringing the roots closer together.
  • A smaller absolute value of $a$ (like $a = 0.5$) makes the parabola wider, spreading the roots apart.

2. The Role of $b$

• Position of the Vertex:
  • The number $b$ affects where the highest or lowest point of the parabola (called the vertex) is located on the x-axis. You can find this position using this formula:

$$x = -\frac{b}{2a}$$

As $b$ changes, the vertex moves left or right, which also affects how the roots are arranged.

• Effect on Roots:
  • Changing $b$ can also move the roots. For example, if $b$ increases while $a$ and $c$ stay the same, the vertex moves, which can change if the parabola crosses the x-axis (meaning it has real roots).

3. The Role of $c$

• Y-Intercept:

  • The value $c$ tells us where the graph hits the y-axis when $x=0$. This influences the vertical position of the parabola.

• Impact on Roots:

  • Changing $c$ can result in different scenarios for the roots:
    • If $c$ is greater than 0, the graph starts above the x-axis (if the vertex is above the x-axis). It could have two, one, or no real roots, which depends on a calculation involving the discriminant: $D = b^2 - 4ac$.
    • If $c$ is less than 0, the parabola starts below the x-axis, which can also lead to different possibilities for the roots.

4. Understanding the Discriminant

• The discriminant, which is $D = b^2 - 4ac$, is key to understanding the roots:
  • If $D > 0$: There are two distinct real roots.
  • If $D = 0$: There is one real root (both roots are the same).
  • If $D < 0$: There are no real roots (the roots are complex).

Conclusion

In short, the numbers $a$, $b$, and $c$ greatly affect the roots of a quadratic equation. They determine how many roots there are, where they are located, and what kind they are, as well as shaping the graph of the quadratic function. Knowing how these parts work together is essential in grade 9 algebra and sets a good foundation for learning more about parabolas and quadratic functions.