The given equation, $y = ax^2 + bx + c$, represents a quadratic equation. In this equation:

• $a$, $b$, and $c$ are constants.
• $x$ is the variable.
• $y$ represents the value of the quadratic expression for a given $x$.

Key features of a quadratic equation:

1. Parabola Shape: The graph of a quadratic equation is a parabola.

   • If $a > 0$, the parabola opens upwards.
   • If $a < 0$, the parabola opens downwards.

2. Vertex: The vertex is the highest or lowest point on the parabola.

   • The vertex $x$-coordinate is found using $x = \frac{-b}{2a}$.

3. Axis of Symmetry: The line $x = \frac{-b}{2a}$ is the axis of symmetry for the parabola.

4. Roots: The solutions to the equation $ax^2 + bx + c = 0$ are called roots or zeros. They can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

5. Discriminant: The term $b^2 - 4ac$ (inside the square root of the quadratic formula) is called the discriminant. It determines the nature of the roots:

   • If $b^2 - 4ac > 0$, there are two distinct real roots.
   • If $b^2 - 4ac = 0$, there is one real repeated root.
   • If $b^2 - 4ac < 0$, there are two complex roots.

Would you like more details on any part of this?

Here are 5 related questions:

1. How do you complete the square for the quadratic equation $y = ax^2 + bx + c$?
2. What is the geometric meaning of the discriminant?
3. How do you derive the quadratic formula?
4. How can you graph a quadratic function from its equation?
5. What is the significance of the vertex form of a quadratic equation?

Tip: To find the vertex of a parabola quickly, use the formula $x = \frac{-b}{2a}$, and substitute that $x$-value into the original equation to find the corresponding $y$-coordinate.