Learning Goal

Syllabus Requirement

Students need to have knowledge of the effect of simple transformations on the graph of ??y = f(x)??, with the following points highlighted:

•    Students should be able to apply one of these transformations to either a quadratic, cubic or reciprocal function and sketch the resulting graph.
•    Given the graph of any function ??y = f(x)?? students should be able to sketch the graph resulting from one of these transformations:
??y = af(x)??
??y = f(x) + a??
??y = f(x + a)??
??y = f(ax)??

Study Notes

There are only 4 simple transformations to learn and the result of these can be split into to two types:

•    Translations of the curve of a function in either the ??x?? or ??y?? axis.
•    Stretching of the curve of a function in either the ??x?? or ??y?? axis.
The result of each of these transformations are listed in the information box below. The same rules can be used for either a quadratic, cubic or reciprocal functions.

The transformations resulting in a translation of the curve of a function ??f(x)?? have an addition to the function

??y = f(x + a)??:
Results in a translation of ??-a?? along the ??x??-axis.
??\implies?? We add ??-a?? to each ??x?? coordinate of the curve, the ??y?? coordinates remain unchanged

??y = f(x) + a??:
Results in a translation of ??+a?? along the ??y??-axis
??\implies?? We add ??+a?? to each ??y?? coordinate of the curve, the ??x?? coordinates remain unchanged

The transformations resulting in a stretch of the curve of a function ??f(x)?? have a multiplication to the function

??y = f(ax)??:
Results in a stretch of ??\frac {1}{a}?? along the ??x??-axis
??\implies?? We multiply each ??x?? coordinate by ??\frac {1}{a}??, the ??y?? coordinates remain unchanged

??y = af(x)??:
Results in a stretch of ??a?? along the ??y??-axis
??\implies?? We multiply each ??y?? coordinate by ??a??, the ??x?? coordinates remain unchanged

Note a simple way of remembering which axis the transformation will be, is if ??a?? is inside or outside the brackets.
If inside the bracket eg. ??y = f(x + a)??, then the transformation will be along the ??x?? axis
If outside the bracket eg ??y = f(ax)??, then the transformation will be along the ??y?? axis

Remember, the same rules can be used for either a quadratic, cubic or reciprocal functions.

Worked Examples

We'll now run through an example of each transformation in turn on the same function:

Figure 1 - Graph of ??y = f(x)??

Figure 1 shows a sketch of the curve with equation ??y = f(x)??.

The curve has a turning point at ??(2, -4)?? and crosses the ??x??-axis at the points ??(0, 0)?? and ??(3, 0)??.

Using     ??y = f(x + a)??

Example 1

Question

Given the sketch of the curve with equation ??y = f(x)?? shown in Figure 1 above.

Sketch the curve with equation ??y = f(x+2)??

On the diagram, show clearly the coordinates of all the points at which the curve meets the axis.

Using     ??y = f(x) + a??

Example 2

Question

Given the sketch of the curve with equation ??y = f(x)?? shown in Figure 1 above.

Sketch the curve with equation ??y = f(x)-3??

On the diagram, show clearly the coordinates of the turning point and intersection with the ??y?? axis.

Using     ??y = f(ax)??

Example 3

Question

Given the sketch of the curve with equation ??y = f(x)?? shown in Figure 1 above.

Sketch the curve with equation ??y = f(2x)??

On the diagram, show clearly the coordinates of all the points at which the curve meets the axis.

Using     ??y = af(x)??

Example 4

Question

Given the sketch of the curve with equation ??y = f(x)?? shown in Figure 1 above.

Sketch the curve with equation ??y = 2f(x)??

On the diagram, show clearly the coordinates of all the points at which the curve meets the axis.

Exam Questions

The table below contains exam questions that have been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Question
Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2005 January Core 1 6 n/a 6
Edexcel Maths Standard 2005 June Core 1 4 n/a 5
Edexcel Maths Standard 2006 January Core 1 6 n/a 9
Edexcel Maths Standard 2006 June Core 1 3 (b) 2
Edexcel Maths Standard 2007 January Core 1 3 n/a 6
Edexcel Maths Standard 2007 June Core 1 5 n/a 5
Edexcel Maths Standard 2008 January Core 1 6 n/a 7
Edexcel Maths Standard 2008 June Core 1 3 n/a 5
Edexcel Maths Standard 2009 January Core 1 5 n/a 6

Exam Tips

1. Exam questions typically take the form shown in the Worked Examples where a function is given and each transformation type tested.

2. The transformations can be grouped into the following two types:
Translation of the curve of a function ??f(x)?? which have an addition to the function.
??y = f(x {\color{Red} +} a)?? or ??y = f(x) {\color{Red} +} a??
Stretching of the curve of a function ??f(x)?? which have a multiplication to the function.
??y = f(ax)?? or ??y = af(x)??

3. A quick way to determine along which axis the the transformation will act is to inspect if ??a?? is inside or outside the bracket:
Inside the bracket the transformation will be along the ??x?? axis
??y = f(x + {\color{Red} a})?? or ??y = f({\color{Red} a}x)??
Outside the bracket the transformation will be along the ??y?? axis
??y = f(x) + {\color{Red} a}?? or ??y = {\color{Red} a}f(x)??

4. Remember that the same rules can be used for either a quadratic, cubic or reciprocal functions.