Transforming graphs

Now, don’t get me wrong, I live in the real world and I completely get that sometimes we just need a way of remembering how to do stuff. But over the past however many years, when I have picked up a year 12 AS maths class, the thing that has annoyed me the most is their understanding of transforming functions from GCSE. They often have no idea what the function notation means, how it works nor how it relates to the graph. This leads to difficulty for some students later on when it comes to composite transformations and getting a real understanding of solving trig equations. I do understand, to a certain extent, that it is usually one of the last few things taught at the end of year 11 and is often rushed as a result but in my opinion, if it’s worth teaching, it’s worth teaching well..

This year, for the first time in a number of years, I have had the opportunity to work with a higher ability year 11 class (I’m usually a C/D bordeline crusader). I have an enthusiastic set 2 of which many plan to do A-Level maths. Having not taught a ‘proper’ higher group for a while, I was anxious about transformations of functions coming up on the scheme of work as I knew I wanted to do it properly, but I was unsure of what approach to take. I had an IT room lesson which I thought would be the perfect opportunity for them to play around with some graphs on Desmos and get a feel for what was going on. I structured the lesson using this worksheet and I had got them to write everything up in their exercise books.


Desmos: A jolly good bit of graphing software

There was a great buzz in the room as they worked and some fantastic discussions could be heard throughout the lesson. It was one of those lessons, where as a teacher, there was very little to do and the students were fully engaged throughout (who doesn’t love using Desmos!). Towards the end of the lesson as I went around the room to check their understanding, I heard someone say those words: “if it’s in the brackets, just do the opposite”. My heart sank a little. As I questioned them further, it became apparent that they had noticed what happens, but had no idea why it was happening. They were able to predict what would happen when I showed them a function and knew what it would look like but their overall understanding was fragile at best. I do believe there was value in the activity, however, I felt that there was still some learning that needed to take place before moving on..

I put together this lesson as an introduction to function notation and tried to develop a strong understanding of why the notation was useful and what it means. I extended it to composite functions, which they seemed to really enjoy!

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The following lesson is where I introduced the idea of function notation and graphs (I used function machines to highlight further what was going on).

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We talked about everything in terms of inputs (x-coordinates) and outputs (y-coordinates). As we looked at each type of transformation, we discussed whether the input or the output was affected and what the significance of this was. They were given mini graphs to draw on for each type of transformation as we went. It was a very ‘teacher led’ lesson – a direct contrast to the IT room lesson which, to be honest, I don’t have a problem with as I know my class respond well to either. My moto is: whatever gets them learning, do that sh*t. As the lesson unfolded, I asked them similar questions to those I asked towards the end of the Desmos lesson and this time I was much more satisfied with their responses. I still think there was a lot of value in the Desmos lesson (I would definitely do it in the same way again), it just needed some good old fashioned teacher led learning (and a good explanation) to go with it.

Here are all the resources mentioned in the above post for those of you who use Smart Notebook. You can find more GCSE graph resources here.

If you have any comments, thoughts or ideas, please share below.


9 thoughts on “Transforming graphs

  1. Permalink  ⋅ Reply

    Paula Beardell Krieg

    April 5, 2015 at 3:13am

    Okay the students didn’t get the understanding from Desmos that you ultimately wanted. I wonder, though, if they better understood what you had to teach them after having worked part of it our for themselves.

    • Permalink  ⋅ Reply


      April 5, 2015 at 9:45am

      Definitely. Which is why I would do it in this way again. Sometimes we want students to just play around with some maths so they can start connecting the dots. My point, in this post was that although they would have been able to answer exam questions on the topic straight after the Desmos lesson, they didn’t really understand why it was working in this way. The follow up lesson helped them to connect the rest of the dots.
      Thanks for reading and commenting 🙂

  2. Permalink  ⋅ Reply

    Paula Beardell Krieg

    April 5, 2015 at 3:09pm

    Thanks for writing up this story. It seems to me that the sequence you describe is a brilliant way of working.

  3. Permalink  ⋅ Reply

    Colleen Young

    April 6, 2015 at 9:36am

    Great resources, thank you. I find that the secret with so many topics, not just Maths is to visit it more than once! More than ever this year I have been revising all through the course and will frequently use a starter or plenary to revisit topics.
    With Desmos particularly for the older students I like the way you can use function notation to plot the graphs. See

    • Permalink  ⋅ Reply


      April 6, 2015 at 10:06am

      I completely agree about revisiting topics. I have been trying out an idea of ‘cumulative starters’ with my year 10s where starters are a set of questions based on things they have done previously. I guess, with the way SOW are usually put together, there often isn’t much opportunity to revisit topics such as graph transformations..
      I honestly hadn’t realised you could use function notation in Desmos in this way – how exciting! I’m looking forward to having a play with this and will definitely try and put together a revision activity for my year 11s when next in the IT room.
      Thanks for reading, commenting and sharing!

  4. Permalink  ⋅ Reply


    October 29, 2019 at 6:28pm

    I’m late to the party but I really wanted to say thanks for these resources. I’ve got a student who is struggling with graph transformations and this way of explaining strikes me as a much more intuitive way to ‘get’ why the transformations work the way they do.

    • Permalink  ⋅ Reply


      October 30, 2019 at 9:56pm

      Pleased to hear it! Thank you for taking the tome to comment 🙂

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