Transformations of Functions

A collection of resources on Transformations of Functions this week. The subject content for GCSE content includes: 13.sketch translations and reflections of a given function and for A Level we have:

Transum Mathematics

From Transum, this wonderful resource, Transformations of Functions shows how various transformations affect the graph of a function. There are 16 possible transformations to try including examples of combinations of transformations. Having chosen your transformation you can try sketching the transformed function before revealing the answer.

Remember that Transum’s index is very helpful, if we look at the Graphs menu, we see numerous activities on graphs, also, on the left hand side activities mapped to a curriculum for graphs.

Mathisfun!

For some helpful notes, check Mathisfun!. Scroll to the bottom of the page for some multiple choice questions to try. The examples include combinations of transformations.

A very helpful feature of the questions on Mathisfun is the very clear feedback, here for example, we see:

Transformations on Desmos

It is very helpful indeed when studying this topic to use graphing software to experiment and try out different functions. Note how easy it is using the Desmos graphing calculator to show a graph and then the same graph after a transformation. For example see here the graph of x2 and (x+a)(or click on  the image).See these Desmos pages – Transformations and Transformations – Advanced.

Transformations of Graphs – Steven Fan on GeoGebra

This is a lovely resource on GeoGebra, Transformation of Graphs from Steven Fan can be used to explore combinations of transformations.

MEI Student Tasks for AS and A Level

Looking at MEI’s student tasks, several are available for exploring transformations with instructions for using the software and questions to explore. Tasks are available for Autograph, Casio, Desmos and GeoGebra.

Further helpful resources:

Dr Frost – Graphs and Transformations

On Dr Frost, GCSE Graph Transformations and for A Level, Graphs and Transformations.for Year 1 and Functions and Graphs for Year 2.

Owen – Functions and Graphs, Year 2

Owen (Owen134866 on TES) has a library of Mathematics Further Mathematics teaching resources, these are really clearly structured with step by step examples. Graphs and Transformations can be found on TES in his A Level Pure Mathematics Year 1/AS Collection and Functions and Graphs which includes combinations of function can be found on TES in his A Level Pure Mathematics Year 2 Collection.

On ck12, 1.7 Function Graphs: Combined Transformations, there are several very useful examples of combining transformations. This content is very clear.

Maths Genie – Transforming Graphs GCSE questions and solutions. Sketching and Transforming Curves A Level questions and solutions.

Corbett Maths Transformations of Graphs practice questions.

On Exam Solutions, we have worked exam questions and helpful video tutorials on Translations of Graphs, reflections and stretches.

Notes from Mathisfun have been mentioned above, further notes include from MathsBitsNotebook.com, this on Transformations of Functions. and the A Level course material from CIMT includes Graph Transforms.

Open Middle

On Open Middle you can now get Google Slides versions of problems – Virtual Activities in Google Slides. The slides have been created by Dan Shuster based on a design by Robert Kaplinsky. Each link is a force-copy link to a Google Slides file. This introduction from Alisha Zare includes ideas for implementing these tasks for students. Note too the webinars available on Open Middle, one for Elementary and one for Secondary.

Virtual Activities in Google Slides

In an earlier post, I looked at this lovely problem, Create a System of Two Equations by Daniel Luevanos on Open Middle, accessible for students yet such a great task for mathematical thinking. We could discuss inequalities here as well as simultaneous equations. This is an 8th grade problem, so found in that collection. Note in that post the Graspable Math canvas and my Desmos page for this problem.

For another wonderful source of resources, from Tim Brzezinski, this brilliant GeoGebra book of Open Middle themed problems. Many problems in the GeoGebra book are exact digital analogues of those found on Open Middle’s site, with other problems characteristic of the Open Middle theme. Do check Tim’s collection of GeoGebra resources.

GeoGebra Book of Open Middle themed problems – Tim Brzezinski

Or perhaps try creating a right -angled triangle:

And we also have from John Rowe, some Open Middle problems as Desmos activities, CL Newsletter October 2020: Open Middle; note the Open Middle template for anyone wishing to design their own activities.

CL Newsletter October 2020 – Open Middle

The Modulus Function

Looking at the Subject content for A Level Mathematics we see that students are expected to be able to understand and use graphs of functions including the modulus of a linear function.

Additionally, students should understand simple transformations of graphs.

This is an ideal opportunity to use graphing technology to understand and explore the modulus (or absolute value) function.

We can use Desmos to illustrate the functions f(x)=2x+3 and g(x)=|2x+3| illustrating that the left portion of g(x) is a reflection of the negative portion of f(x). We could also explore translations parallel to the x and y axes.

Looking at MEI’s excellent student tasks for A Level, we see that the a Modulus Function task is provided for Autograph, the Casio CG50 graphing calculator, Desmos and GeoGebra.

Like all the MEI student tasks instructions for the use of the software are provided some great questions to get students thinking and exploring.

Edexcel’s guide to using GeoGebra for AS and A Level Mathematics includes several GeoGebra Interactives for students to explore.

Edexcel – Guide to using GeoGebra

This on exploring the intersection of lines and modulus graphs is very useful for solving equations involving modulus functions.

Edexcel – guide to GeoGebra for A Level Mathematics

There are several sources of notes and examples as well as worked exam questions on this topic. Explore these below – a reminder of the many excellent sites available.

Of course we have to start with the outstanding Dr Frost Maths, (you might have seen him in the news recently!) where you will find resources on the Modulus Function under Graphs and Functions for Key Stage 5. Scroll down for “Sketch the modulus of a linear function.”

Dr Frost Maths – Modulus Functions

Owen (Owen134866 on TES) has a library of Mathematics Further Mathematics teaching resources, these are really clearly structured with step by step examples. Functions and Graphs includes the modulus function and can be found on TES in his A Level Pure Mathematics Year 2 Collection.


Exam Solutions

On Exam Solutions we have four worked examination questions on Modulus Inequalities, additionally there are two helpful tutorials, one is an introduction to the modulus function and the other on Graphing y=|f(x)|.


HELM Notes – Some Common Functions

One of the series of excellent HELM Notes is Some Common Functions under Section 2 on Basic Functions; the modulus function is briefly covered in Section 3 of the document, a worked example is provided to plot a modulus function.


Maths Genie – Functions

On Maths Genie the questions and solutions on Functions includes one question (7) on the Modulus function. Students should be able to solve problems such as finding the values of x which satisfy │2x – 3│< 9; they should be able to do this both algebraically and graphically.


From the Math Centre, we have a first aid kit on the modulus symbol.

Use of Technology

We see from the A Level subject content that the use of technology must permeate the study of AS and A level mathematics. I believe that is true for younger students also; technology can help us make explanations very clear and allow us to show multiple representations of a concept.

At Maths Conference #24 I showed many examples of resources to use in class, the following list gives links to those resources. Many of the links here take you to a post with further information on the resource.

Standards Unit

Many of the tasks refer to software which no longer works on modern browsers, happily you can still access much of this from this treasure trove – numworx.

Transition to A Level Mathematics – amsp

On the subject of graph sketching – teach everybody to do this from a young age and use graphing software all the time to show students the links between algebra and graphical representations. For year 11 and 12 students, from The Advanced Mathematics Support Programme these transitions to A Level Mathematics resources are excellent and include a unit on graph sketching.

Math Warehouse – Quadratic Formula Calculator

The MEI AS/A level Student tasks are excellent, they provide instructions for the use of the software and ask some great questions to get students thinking and exploring. Have a look at this Autograph task on Numerical Methods – Fixed Point Iteration. Autograph is brilliant for work on Iterative processes – the task shows how quick it is to use. You can zoom in on the image as you can see here. Remember that Autograph is now completely free. There is a wealth of information on how to use it available on TSM Resources.

Transum Mathematics – Refreshing Revision

This Custom Starter from Transum, Refreshing Revision, allows teachers to select the number of questions and the topics to include; scroll down the page and choose the topics you want from the Concept Selection. It is possible to save a particular selection of topics as the URL for your selection will be generated. It is also possible to drag the panels so your questions are displayed in the desired order. The beginning of a lesson can be an ideal time to review previous learning, starters like this can be ideal.

This post on Transum includes details of some other resources on the site, note the very helpful index for teachers and another for students. See also Transum – Advanced Starters.

Transum – Advanced Starters

Maths at school … and at home

Both White Rose Maths and BBC Bitesize provided Daily Lessons for students studying at home earlier this year and both are continuing to support students studying at home as well as in school.

If we look at Home Learning from White Rose Maths we see that they are continuing to produce excellent daily ‘home learning’ lessons for Years 1 to 9 (UK ages 6 to 14). Each lesson has a very clear video to help students complete the activity.

White Rose Maths – Daily Lessons

I had a close look at a White Rose lesson on Perimeter in this post; note that you can still see the Summer Term Archive so can access earlier lessons. This particular lesson came from Year 7, Week 5 (scroll down for earlier weeks).

Remember that Nicola Whiston has a superb collection of Knowledge Organisers which follow the White Rose Schemes of Learning, she is sharing the collection here, via Dropbox. My post on Knowledge Organisers has proved very popular, note the new Featured Posts menu on the right to make this easier to find.

From BBC Bitesize, new for this Autumn term are the collections of hand-picked videos, articles, games and quizzes for primary and secondary students, also available are new Primary Catch-Up Lessons. If we look at ‘This Term’s Topics‘ we can then choose a Year and then a subject. Choosing Year 7 Maths for example leads to this collection which includes many of the daily lessons as well as other resources. Note that new topics and lessons are being added to these resources. For an example of a Daily lesson, check this on Perimeter mentioned in the post mentioned earlier on the White Rose and BBC resources. Note the additional resources including a comprehensive worksheet from Pearson for which answers are provided.

BBC Daily Lesson – Perimeter

More valuable resources can be found on the new NCETM website. Check the section for the new school year which provides a range of lesson-planning and professional development resources for primary and secondary teachers. Further resources will be added during the term. The resources here for example concentrate on key mathematical ideas important for pupils whose summer term in Year 6 was disrupted by the lockdown.

amsp transition materials – Sketching

On the subject of Transition, at the other end of secondary education, a reminder of the excellent resources from amsp designed for students to make the transition from GCSE to AS and A level Mathematics. Six sets of resources are available, each set includes either written worked solutions, video solutions or links to websites.