**Syllabus Requirement**

Students need to learn and understand

**Quadratic functions and their graphs.**

A **quadratic function** is a polynomial of the form ??ax^{2} + bx + c = 0?? where ??a??, ??b?? and ??c?? are real numbers.

Note that ??b?? and ??c?? can be equal to 0 but ??a?? must be non-zero for the function to be quadratic.

**The General Shape of Graphs of Quadratic Functions**

Every quadratic function will have a graph that is a parabola, or in simpler words, a "u-shape". The figure below shows the general shape of a quadratic function where ??a?? is positive:

When the ??a?? term is negative, the shape of the parabola is flipped to be upside-down as shown in the figure below:

The effect of changes in the coefficients ??a??, ??b?? and ??c?? on the shape and position of the graph are discussed in a later part of the study notes:

Core 1: 1. Algebra and Functions: j. Knowledge of the Effect of Simple Transformations on the Graph of ??y=f(x)??.

Typically this part of the syllabus is not examined without reference to other areas of the syllabus (i.e. the exam will normally ask that you factorise a quadratic equation, or sketch one, etc.) which will be covered separately. For that reason there is only 1 example on this section:

**Quadratic Functions**

**Example 1**

**Question**

Simplify the following expression ??2x^{2} + 4x + 2 = 2x^{2} - x + 3?? and determine whether it is a quadratic or not.

Show Solution

Typically we cover every exam question that has been asked on this topic however this part of the syllabus is not examined without reference to other areas of the syllabus.

The exam will typically ask that you factorise a quadratic equation, or sketch the graph, which are covered in the following sections. Follow these links to see how we can develop our knowledge of the quadratic function.