This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

Which curve is which, and how would you plan a route to pass between them?

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

Can you massage the parameters of these curves to make them match as closely as possible?

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Compares the size of functions f(n) for large values of n.

Sketch the graphs for this implicitly defined family of functions.

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Can you work out the equations of the trig graphs I used to make my pattern?

Can you sketch these difficult curves, which have uses in mathematical modelling?

In this trail, a new type of NRICH resource, learn about transformations of graphs. Given patterns made from families of graphs find all the equations in the family.

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

Can you work out which processes are represented by the graphs?

Explore how can changing the axes for a plot of an equation can lead to different shaped graphs emerging

Which line graph, equations and physical processes go together?

Explore the rates of growth of the sorts of simple polynomials often used in mathematical modelling.

Explore displacement/time and velocity/time graphs with this mouse motion sensor.