2.1 Maximum / Minimum of Quadratics

Cambridge O Level Additional Mathematics (4037)

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Part 1 — What Is a Quadratic Function?

A quadratic function is any function of the form:

$f(x) = ax^2 + bx + c$

where:

• a, b, and c are numbers (called constants)
• a ≠ 0 (if a were zero, there would be no x² term and it would not be quadratic)

When you draw a quadratic function on a graph, you always get a smooth, curved shape called a parabola (say: pa-RAB-oh-la). Think of it like the shape of a satellite dish or the path of a ball thrown through the air.

Part 2 — The Shape of the Parabola: Which Way Does It Open?

The value of a (the number in front of x²) controls the shape:

Part 3 — What Is a Maximum or Minimum Point?

Every parabola has exactly one special point at its tip — this is called the turning point (also called a stationary point). It is the point where the curve changes direction.

• If the parabola opens upward (U-shape), the turning point is at the bottom → this is the minimum point → it gives the minimum value of f(x).
• If the parabola opens downward (∩-shape), the turning point is at the top → this is the maximum point → it gives the maximum value of f(x).

The turning point has coordinates (h, k), where:

• h is the x-value at the turning point
• k is the maximum or minimum value of the function

Every parabola also has a line of symmetry — an invisible vertical line that passes exactly through the turning point, cutting the parabola into two mirror-image halves. Its equation is always x = h.