# transformations of graphs

► syllabus content for the Topic: SL syllabus (see syllabus section 2.3); HL syllabus (see syllabus section 2.3)

It's very important for students to have a complete and fluent understanding of the following types of transformations that can be applied to the graph of a function:
translations - horizontal & vertical
reflections - in both x- and y-axes
dilations (stretching & shrinking) - in both horizontal and vertical directions

Transformations of graphs is a topic/skil that will be taught early in the course during a unit on functions and then will appear again during other times in the course - especially when studying the graphs of trigonometric functions. Below is a GeoGebra applet to explore transformations applied to the graph of the sine curve.

For some interactive practice on transformations of graphs, go to the page Transformations - interactive practice that contains a dynamic Geogebra applet that automatically checks responses. More applets will be added in the near future.

Some questions involve composite transformations where there is more than transformation to be performed.  Sometimes the order in which the transformations are performed makes a difference and sometimes the order does not make a difference. I've made a one-page document with an example illustrating a situation where it does make a difference. Open the following document: sequence of tranformations of a graph.

It is also useful - especially for HL students - to be familiar with the relationship between the graph of and the graphs of , and $y=\frac{1}{f\left(x\right)}$ .  See 'curve sketching' below - and also see the Geogebra applet on the page Other changes to a graph.

Additionally, I firmly believe that the topic which is most suitable for teaching by means of activities that students carry out themselves (usually in the classroom) is the topic of transformations of graphs. My experience has been that students can gain a very clear understanding of different graph transformations by working through an activity using graphing software (either on a GDC or a computer) that presents the material in a dynamic and interactive manner. In fact, I find that it takes significantly less time to teach the topic using individual technology-based activities with students compared to a traditional lecture along with static information presented in a paper textbook. See my GDC activities in the course planning / teaching notes section further below on this page.

#### Curve sketching (not really transformations of graphs)

While teaching transformations of graphs (translations, reflections & dilations), I think it's a good idea to give students experience with other types of graph 'changes' (see Other changes to a graph).  I put this in the general category of curve sketching.  For example, when students are studying differential calculus they should be able to sketch the graph of a function given the graph of the derivative of the function (and a point or two to 'anchor' it).  Before that students should be exposed to questions like the one below (click on the 'eye').  'Curve sketching' questions appear more often on HL exams than on SL exams.  Nevertheless, questions like these require good conceptual understanding and a willingness to think logically - good for all students to learn and practice. Most students find these kinds of questions challenging - where there is no computation involved, just logic and understanding.  In recent years, questions like these seem to be appearing more often on Paper 1 exams for Maths HL & SL exams.

'Curve sketching' question (link to solution provided)

### 4 questions - ‘accessible’ to ‘discriminating’

accessible SL question

moderate SL / accessible HL question

discriminating SL / moderate HL question

discriminating HL question

### Course planning / teaching notes:

Consider teaching transformations of graphs by having your students individually work through an activity using graphing software - either on their GDC or with computer software. It's best if there is some dynamic aspects to the graphing utility - for example, creating sliders that will change a particular parameter in a function that then simultaneously changes the graph of that function. Below are two images from a 7-page activity that I wrote for the TI-Nspire handheld device (GDC) that uses dynamic sliders. I hope to soon be adding a similar activity for the TI-84 GDC. Click on either of the images to obtain a PDF copy of the activity that is also listed in the teaching materials section below.

#### Exam-style question on transformations of graphs

On the sample mock exam MockA_SL_Paper_1_2014_v1, question 10 part B involved transformations of graphs.  The last part of the question (sub-part c) is tricky for SL students (and HL).  To see the question and some explanatory notes on sub-parts b and c, open the following document: MockA_SL_P1_2014_Q10_Part_B.  Also see the notes on sequence of transformations of a graph (presented above) that has an example illustrating some guidelines on the sequential order of transformations in a composite transformation.

♦ teaching materials

EXS_2-3-40v2_SLHL_transformations_graphs
A set of 7 questions covering the full range of transformations of graphs - translations, reflections & dilations (answers included) - corrected version 2

ACT_2-3-30v3_SLHL_transformations_graphs_Nspire
An activity that users sliders and dynamic graphs to teach transformations of graphs; students will answer a series of questions/conjectures based on their observations; suitable for beginners to the TI-Nspire - contains thorough instructions supported by images (7 pages)

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