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.

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

Can you make a curve to match my friend's requirements?

If you plot these graphs they may look the same, but are they?

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

Sketch the graphs for this implicitly defined family of functions.

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

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

Which line graph, equations and physical processes go together?

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

By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

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

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

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

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.

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

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

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

What biological growth processes can you fit to these graphs?

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

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.

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?

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

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

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.

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