“Teach A Level Maths” Vol. 1: AS Core Modules 20: Stretches © Christine Crisp

Module C1 Module C2 Edexcel AQA OCR MEI/OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

We have seen that graphs can be translated. e.g. The translation of the function by the vector gives the function . The graph becomes We will now look at other transformations.

e.g.1 Consider the following functions: and For For In transforming from to the y-value has been multiplied by 4

e.g.1 Consider the following functions: and For For In transforming from to the y-value has been multiplied by 4 Similarly, for every value of x, the y-value on is 4 times the y-value on is a stretch of scale factor 4 parallel to the y-axis

BUT, you may look at the graph and see the transformation differently. The graphs of the functions are as follows: is a stretch of by scale factor 4, parallel to the y-axis BUT, you may look at the graph and see the transformation differently.

has been squashed in the x-direction We say there is a stretch of scale factor parallel to the x-axis.

is a transformation of given by either a stretch of scale factor 4 parallel to the y-axis or a stretch of scale factor parallel to the x-axis

It is easier to see the value of the stretch in the y direction. To obtain from we multiply every value of y by 4. The reason for the size of the 2nd stretch can be seen more easily if we write as Now, for 2 and for 1 The x-value must be halved to give the same value of y.

The x-value must be halved to give the same value of y. It is easier to see the value of the stretch in the y direction. To obtain from we multiply every value of y by 4. The reason for the size of the 2nd stretch can be seen more easily if we write as Now, for 2 and for 1 The x-value must be halved to give the same value of y.

SUMMARY The transformation of to is a stretch of scale factor 4 parallel to the y-axis or is a stretch of scale factor parallel to the x-axis

SUMMARY The function is obtained from by a stretch of scale factor ( s.f. ) k, parallel to the y-axis. The function is obtained from by a stretch of scale factor ( s.f. ) , parallel to the x-axis.

e.g. 2 Describe the transformation of that gives . Using the same axes, sketch both functions. Solution: can be written as so it is a stretch of s.f. 3, parallel to the y-axis We always stretch from an axis.

Exercises 1. (a) Describe a transformation of that gives . (b) Sketch the graphs of both functions to illustrate your answer. Solution: (a) A stretch of s.f. 9 parallel to the y-axis. OR A stretch of s.f. parallel to the x-axis. ( The 1st of these is easier, especially if we have, for example ) (b)

Exercises 2. The sketch below shows a function . Copy the sketch and, using a new set of axes for each, sketch the following, labelling the axes clearly: (a) (b) Describe each transformation in words.

Solution: (a) (b) Stretch, s.f. parallel to the x-axis Stretch, s.f. parallel to the y-axis

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

SUMMARY The function is obtained from by a stretch of scale factor ( s.f. ) , parallel to the x-axis. by a stretch of scale factor ( s.f. ) k, parallel to the y-axis.

is a transformation of given by either a stretch of scale factor 4 parallel to the y-axis or a stretch of scale factor parallel to the x-axis e.g. 1

We always stretch from an axis. Using the same axes, sketch both functions. so it is a stretch of s.f. 3, parallel to the y-axis e.g. 2 Describe the transformation of that gives . Solution: can be written as