# Patterns and Relationships: Level 6

The key idea of patterns and relationships at level 6 is that graphs, tables and equations are alternative representations of functions and that transformations of functions have predictable effects on these representations.

At level 6 the focus is on linear and quadratic functions. For linear functions students need to be familiar with how *x *and *y* intercepts and gradients of graphs are related to the equation representation. This provides the foundation for the key idea of calculus at level 7 and also for describing graphs of quadratic equations in terms of roots, *y* intercept and axis of symmetry. The material in this thread therefore has strong connections with the key idea of Equations and Expressions at Level 6 as solutions of equations provide the roots. Real life contexts for functions provide meaning to roots as solutions to problems, gradients as rates of change and vertices as minimum/maximum values.

An important concept to understand at this level is transforming equations by: adding/subtracting a constant to the function of *x* to move the graph up/down the *y* axis; adding/subtracting a constant to x within the function to move the graph down/up the *x* axis; and multiplying *x* by a constant to alter the gradient. Graphics calculators provide a powerful tool for exploring these transformations of equations as general principles that apply to linear, quadratic and other functions rather than as specific rules for quadratic equations. Manipulation of algebraic expressions is also important for transformations and so generalising the properties of operations from numbers to variables is also a key component of this thread.

This key idea develops from the key idea of patterns and relationships at level 5, where students are beginning to understand the relationship between graphs and equations.

This key idea is extended to the key idea of patterns and relationships at level 7 where the students apply formulae to real world problems.