AM
Animated Mathematics

Quadratic Equations: Content

Introduction

An introduction to quadratic equations and rules for simplifying quadratic equations.

Graphing Quadratic Equations

Graphing a quadratic equation by plotting many points on the line. All quadratic equations have an axis of mirror symmetry and a minimum or maximum point.

Quadratic Transformations

We discuss the transformations, vertical shift, reflection, and stretch.

The form y=ax²+c

The solutions, axis of symmetry, and vertex of a quadratic equation where b=0.

The y=ax²+bx

The solutions can be found by first factorising the equation.

Factorisation

Factorising a quadratic equation explained geometrically.

Graphing y=(x+n)(x+m)

Finding the solutions, axis of symmetry, and vertex of a factorisable quadratic equation.

Designing a Quadratic Formula

We discuss how to design a quadratic equation by placing the solutions and vertex.

A Perfect Square

Perfect quadratic squares have equals solutions and the curve touches the x-axis

Transformations II

Transformation of a quadratic equation such as stretch, reflection, vertical and horizontal displacment are explored.

Completing the Square

We discuss completing the square for quadratic equations to find the solutions.

Completing the Square and Transformation

Completing the square shows that all quadratic curves can be formed by applying transformation to the simplest quadratic curve y=x²

The Quadratic Formula

The quadratic formula is derived step by step by completing the square of the quadratic equation y=ax²+bx+c. This sections does not contain any exercises.

Quadratic Formula and Discriminant

The quadratic formula can be used to find the solutions of any quadratic equation. The number of solutions is determined by the sign of the descriminant. A quadratic graph with no solutions does not cross the x-axis, and graphs with one solution touches the x-axis at one point.