Parabolas (quadratic graphs)

What a parabola is

• A parabola is the graph of a quadratic function — one with an $x^2$ term as its highest power.
• Its equation can be written in a few useful forms:
  • General form $y = ax^2 + bx + c$.
  • Factored form $y = a(x - p)(x - q)$ — best for reading the roots $p$ and $q$.
  • Vertex form $y = a(x - h)^2 + k$ — best for reading the vertex $(h, k)$.

What the number $a$ does

• $a$ controls the direction and width:
  • $a > 0$ ⇒ the parabola opens upward (a "smile", with a minimum).
  • $a < 0$ ⇒ it opens downward (a "frown", with a maximum).
  • The larger $|a|$, the narrower the parabola; a small $|a|$ makes it wide.

Finding the vertex

• The vertex sits on the axis of symmetry, exactly halfway between the roots:

$x = \frac{p + q}{2}$

• Substitute that $x$ back into the equation to get the vertex's $y$-value.
• From vertex form $y = a(x - h)^2 + k$ you can read the vertex directly as $(h, k)$.

Sketching a parabola

• Find the $y$-intercept (set $x = 0$).
• Find the $x$-intercepts / roots (set $y = 0$ and factorise, or use the quadratic formula).
• Find the vertex (midpoint of the roots, then substitute).
• Decide the direction from the sign of $a$, then draw a smooth symmetric curve through the points.