# Quadratic Functions

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths 0606
> Source: https://www.owlsprep.com/study/cie-0606-u2-overview/
> Weight: A high-frequency topic that appears on both Paper 1 and Paper 2

This unit builds on basic quadratic equation skills to explore core properties of quadratic functions, including vertex values, root classification, and quadratic inequality solutions, critical for algebra and modelling exam questions.

**Prerequisites:** Basic quadratic factoring and linear equation solving

## Learning objectives

- Rewrite quadratic expressions in completed square form to identify vertex coordinates, maximum/minimum values, and function ranges
- Use the discriminant to classify roots of quadratic equations and solve problems involving equal, real distinct, or no real roots
- Solve linear and quadratic inequalities using both algebraic and graphical interpretation methods
- Apply quadratic function properties to solve real-world modelling and exam-style problems

## Unit at a glance

The unit follows a logical sequence that connects algebraic manipulation to graphical interpretation of quadratics, a key cross-cutting skill for the rest of the syllabus. You will first master completing the square, a foundational technique that lets you quickly find the maximum or minimum output of any quadratic, as well as its full range for a given domain.

You will then move to the discriminant rule, which lets you classify the number and type of roots a quadratic equation has without solving it fully, before combining your algebra and graph knowledge to solve quadratic inequalities and state valid solution ranges.

Work through the following sub-topics in order to build mastery:
- [Completing the Square, Max/Min and Range](https://www.owlsprep.com/study/cie-0606-u2-completing-the-square-max-min/) — Learn to rewrite quadratic expressions in completed square form to identify vertex coordinates, maximum/minimum values, and calculate function ranges for specified domains.
- [Discriminant, Roots and Quadratic Inequalities](https://www.owlsprep.com/study/cie-0606-u2-discriminant-roots-and-quadratic-inequalities/) — Use the discriminant to classify root types, solve problems involving root conditions, and solve one- and two-sided quadratic inequalities using algebraic and graphical methods.

## Common pitfalls

- **Wrong:** Forgetting to flip the inequality sign when multiplying/dividing both sides of a quadratic inequality by a negative number
  - Why it fails: This reverses the direction of valid solutions, leading to incorrect answer ranges that lose marks in exams
  - Correct: Always reverse the inequality symbol if you multiply or divide across the inequality by a negative value, or use graphical interpretation to verify your solution range
- **Wrong:** Calculating the discriminant as $b^2 + 4ac$ instead of $b^2 - 4ac$
  - Why it fails: The sign of the discriminant determines root type, so this sign error completely changes your classification of roots
  - Correct: Memorise the discriminant formula as $Δ = b^2 - 4ac$, and double-check the sign of the $ac$ term before subtracting
- **Wrong:** Assuming the vertex of $ax^2 + bx + c$ is always a minimum value
  - Why it fails: The sign of coefficient $a$ determines if the parabola opens upwards (minimum vertex) or downwards (maximum vertex)
  - Correct: Check the $x^2$ coefficient first: positive $a$ = minimum vertex, negative $a$ = maximum vertex

## Cheatsheet

| Concept | Formula/Rule | Sub-topic Reference |
| --- | --- | --- |
| Completed Square Form | $a(x + h)^2 + k$ | Completing the Square, Max/Min and Range |
| Vertex Coordinates | $(-h, k)$ for $a(x+h)^2 +k$ | Completing the Square, Max/Min and Range |
| Discriminant | $Δ = b^2 - 4ac$ | Discriminant, Roots and Quadratic Inequalities |
| Equal Roots Condition | $Δ = 0$ | Discriminant, Roots and Quadratic Inequalities |
| Real Distinct Roots Condition | $Δ > 0$ | Discriminant, Roots and Quadratic Inequalities |
| No Real Roots Condition | $Δ < 0$ | Discriminant, Roots and Quadratic Inequalities |

## What's next

Start your study of this unit with the first sub-topic on completing the square, where you will learn how to rearrange quadratic expressions to quickly find their turning points and range, a skill frequently tested in both short and long exam questions. Once you have mastered all content in this Quadratic Functions unit, you will move on to the next unit covering Factors of Polynomials, which builds directly on the algebraic manipulation skills you practice here.

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