Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

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

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

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

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

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

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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.

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

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

Sketch the graphs for this implicitly defined family of functions.

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?

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

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

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

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 draw the height-time chart as this complicated vessel fills with water?

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

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 work out which processes are represented by the graphs?

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

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

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?

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.

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?

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

What biological growth processes can you fit to these graphs?

Can you match the charts of these functions to the charts of their integrals?