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Broad Topics > Sequences, Functions and Graphs > Graphs

Exponential Trend

Stage: 5 Challenge Level:

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.

Stage: 4 Challenge Level:

Four vehicles travelled on a road. What can you deduce from the times that they met?

Mathsjam Jars

Stage: 4 Challenge Level:

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

Cubics

Stage: 4 and 5 Challenge Level:

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

Steve's Mapping

Stage: 5 Challenge Level:

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

Stage: 4 Challenge Level:

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

Perpendicular Lines

Stage: 4 Challenge Level:

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

Gosh Cosh

Stage: 5 Challenge Level:

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

Parabolas Again

Stage: 4 and 5 Challenge Level:

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

Stage: 4 Challenge Level:

Can you adjust the curve so the bead drops with near constant vertical velocity?

Small Steps

Stage: 5 Challenge Level:

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Which Is Bigger?

Stage: 4 Challenge Level:

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Motion Sensor

Stage: 4 Challenge Level:

Looking at the graph - when was the person moving fastest? Slowest?

Real(ly) Numbers

Stage: 5 Challenge Level:

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Stage: 4 Challenge Level:

Here are some more quadratic functions to explore. How are their graphs related?

Parabolic Patterns

Stage: 4 and 5 Challenge Level:

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.

More Parabolic Patterns

Stage: 4 and 5 Challenge Level:

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.

Stage: 4 Challenge Level:

Explore the two quadratic functions and find out how their graphs are related.

Ellipses

Stage: 4 and 5 Challenge Level:

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

Curve Match

Stage: 5 Challenge Level:

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

Surprising Transformations

Stage: 4 Challenge Level:

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

Maths Filler 2

Stage: 4 Challenge Level:

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

Alison's Mapping

Stage: 4 Challenge Level:

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

Curve Fitter

Stage: 5 Challenge Level:

Can you fit a cubic equation to this graph?

Guess the Function

Stage: 5 Challenge Level:

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

Immersion

Stage: 4 Challenge Level:

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

Stage: 4 Challenge Level:

Explore the relationship between quadratic functions and their graphs.

Equation Matcher

Stage: 5 Challenge Level:

Can you match these equations to these graphs?

Climbing

Stage: 5 Challenge Level:

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

Cubic Spin

Stage: 5 Challenge Level:

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

Parabella

Stage: 5 Challenge Level:

This is a beautiful result involving a parabola and parallels.

After Thought

Stage: 5 Challenge Level:

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Matchless

Stage: 4 Challenge Level:

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Which Is Cheaper?

Stage: 4 Challenge Level:

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

Sangaku

Stage: 5 Challenge Level:

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

Golden Construction

Stage: 5 Challenge Level:

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Graphical Interpretation

Stage: 4 Challenge Level:

This set of resources for teachers offers interactive environments to support work on graphical interpretation at Key Stage 4.

Bio Graphs

Stage: 4 Challenge Level:

What biological growth processes can you fit to these graphs?

Power Up

Stage: 5 Challenge Level:

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

Three Ways

Stage: 5 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Electric Kettle

Stage: 4 Challenge Level:

Explore the relationship between resistance and temperature

Without Calculus

Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

What's That Graph?

Stage: 4 Challenge Level:

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

Interpolating Polynomials

Stage: 5 Challenge Level:

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.

Bus Stop

Stage: 4 Challenge Level:

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

Lap Times

Stage: 4 Challenge Level:

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

Real-life Equations

Stage: 5 Challenge Level:

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Graphic Biology

Stage: 5 Challenge Level:

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