Quadratic and Cubic Graphs

Shape

• $a > 0$: U-shape (minimum point).
• $a < 0$: ∩-shape (maximum point).

Key Features

• Roots (x-intercepts): where $y = 0$.
• Y-intercept: the constant $c$.
• Turning point (vertex): the minimum or maximum.
• Line of symmetry: passes through the vertex, $x = -\frac{b}{2a}$.

Finding the Turning Point

Complete the square: $y = a(x - h)^2 + k$ → vertex at $(h, k)$.

$y = x^2 - 6x + 11 = (x - 3)^2 + 2$ → vertex $(3, 2)$.

Shape

• $a > 0$: rises from bottom-left to top-right.
• $a < 0$: falls from top-left to bottom-right.

Key Features

• Up to 3 roots.
• Up to 2 turning points.

1. Make a table of values.
2. Plot points carefully.
3. Join with a smooth curve (not straight lines).