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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 2

Quadratic functions

Move between algebraic forms, roots and graph features to choose the shortest valid method.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • complete the square and find a turning point
  • determine a range for a stated domain
  • use the discriminant for roots and intersections
  • solve quadratic equations and inequalities

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Move between algebraic forms, roots and graph features to choose the shortest valid method.

A quadratic can be read in three useful ways: expanded form highlights coefficients, factorised form shows roots, and completed-square form shows the turning point.

The discriminant is more than a root test. After substituting a line into a curve, it identifies a chord, tangent or non-intersection without solving fully.

01

SECTION 01

Forms and turning points

Core concept

Completing the square rewrites ax² + bx + c as a(x − h)² + k. The turning point is (h, k); it is a minimum when a > 0 and a maximum when a < 0.

RULE 1
x-coordinate of vertex = −b/(2a)
RULE 2
f(x) = a(x − h)² + k
Original worked example

Expose the minimum value

  1. Start with f(x) = 2x² − 8x + 11.
  2. Factor 2 from the x terms: 2(x² − 4x) + 11.
  3. Complete the square: 2[(x − 2)² − 4] + 11.
  4. Simplify to 2(x − 2)² + 3.

Answer: Minimum value 3 at x = 2; turning point (2, 3).

02

SECTION 02

Graphs, domains and ranges

Core concept

Sketch using the direction of opening, turning point, intercepts and symmetry. A restricted domain may mean the turning point is not included, so evaluate endpoints and consider whether each boundary is open or closed.

DETAILED EXPLANATION

  • The y-intercept is c.
  • Roots are x-intercepts and may be absent.
  • Use ≤ or ≥ when the endpoint is included.
Typical linear, quadratic, cubic, reciprocal and exponential graph shapeslinearquadraticcubicreciprocalexponential
Original graph-family guide: use intercepts, symmetry, turning points and asymptotes to identify each family.
03

SECTION 03

Discriminants and intersections

Core concept

For ax² + bx + c = 0, Δ = b² − 4ac decides the number of real roots. The same test works after combining a line and a curve into one quadratic.

RULE 1
Δ = b² − 4ac
RULE 2
x = (−b ± √Δ)/(2a)
Meaning of the discriminant
ConditionRootsLine and curve
Δ > 0two distinct real rootstwo intersections
Δ = 0one repeated roottangent
Δ < 0no real rootsno intersection
04

SECTION 04

Equations and inequalities

Core concept

Choose factorisation for simple integer roots, completing the square for exact structure, or the formula for a general quadratic. For an inequality, identify the roots and use the sign of the parabola in each interval.

ORIGINAL STUDY DIAGRAMSolve a quadratic inequality
1Move everything to one side
2Find the boundary roots
3Sketch the sign of the parabola
4Select required intervals
5Use correct endpoint symbols

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • Completed-square form reveals the turning point and extreme value.
  • A restricted domain must be considered when finding the range.
  • The discriminant classifies roots, intersections and tangency.
  • Quadratic equations allow several valid methods.
  • Quadratic inequality answers are intervals, not isolated roots.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • complete the square and find a turning point
  • determine a range for a stated domain
  • use the discriminant for roots and intersections
  • solve quadratic equations and inequalities
← FunctionsFactors of polynomials →