Learning Goal

Syllabus Requirement

Students need to learn and understand Quadratic functions and their graphs.

Study Notes

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: The graph of ??y=x^{2}??.

When the ??a?? term is negative, the shape of the parabola is flipped to be upside-down as shown in the figure below: The graph of ??y=-x^{2}??.

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)??.

Worked Examples

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: