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

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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.

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?

A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students.

Can you find trig graphs to satisfy a variety of conditions?

Draw graphs of the sine and modulus functions and explain the humps.

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Be playful with graphs and networks, and see what theorems you can discover!

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

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 can sketch y=f(x), there are several related functions you can also sketch...

What graphs can you make by transforming sine, cosine and tangent graphs?

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.

Look for the common features in these graphs. Which graphs belong together?

You can get a great feel for functions by sketching their graphs or using graph plotting software...

Use graphs to gain insights into an area and perimeter problem, or use your knowledge of area and perimeter to gain insights into the 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.

Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

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

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

How does the temperature of a cup of tea behave over time? What is the radius of a spherical balloon as it is inflated? What is the distance fallen by a parachutist after jumping out of a plane?. . . .

What can you say about this graph? A number of questions have been suggested to help you look at the graph in different ways. Use these to help you make sense of this and similar graphs.

Can you find a quadratic equation which passes close to these points?

Problems to help you get started with higher mathematics courses

Can you describe what an asymptote is? This resource includes a list of statements about asymptotes and a collection of graphs, some of which have asymptotes. Use the graphs to help you decide. . . .

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

Given probabilities of taking paths in a graph from each node, use matrix multiplication to find the probability of going from one vertex to another in 2 stages, or 3, or 4 or even 100.

Solve these differential equations to see how a minus sign can change the answer

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...

Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.

Investigate the mathematics behind blood buffers and derive the form of a titration curve.

Try our new CMEP tasks to help you to think functionally!

Choose some complex numbers and mark them by points on a graph. Multiply your numbers by i once, twice, three times, four times, ..., n times? What happens?

Some graphs grow so quickly you can reach dizzy heights...

Steve has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Can you solve the clues to find out who's who on the friendship graph?

Explore and sketch curves with different properties.

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.

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