# transformations of graphs

** Quick links**:

► downloadable teaching materials for 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.