Definition of Quadratic Equations: Quadratic equations contain an $x^2$ term; the standard form is $y = ax^2 + bx + c$ (where $a \neq 0$).

Role of Coefficient $a$: The sign of $a$ affects the parabola’s direction. If $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards.

Effect of $|a|$ on Width: The larger the absolute value of $a$, the narrower the parabola. The smaller the absolute value of $a$, the wider the parabola.

Vertex and Symmetry: For $y = ax^2$, the vertex is at $(0,0)$, and the parabola is symmetric about the $y$-axis because $x^2$ and $(-x)^2$ yield the same $y$-value.

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