7: More Graphs and Translations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core...

7: More Graphs and 7: More Graphs and TranslationsTranslations

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

More Graphs and Translations

Module C1

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When sketching a graph, we try not to PLOT points. We want the general shape not an accurate drawing.

x

The function is an example of a cubic function.

3xy

• x = 0 y = 0, so the graph goes through the origin.

• As x increases, y increases quickly

e.g. x = 1 y = 1; x = 2 y = 8

To sketch the graph we notice the following:

• The graph has 180 rotational symmetry about the origine.g. x = -1 y = -1;

x = -2 y = -8


3xy 23 3 )(xy

In a similar way, is a translation

of

23 3 )(xy3xy

3

2

We have seen that the quadratic function

is a translation of by23 2 )(xy 2xy

2

3


is another cubic function

xxxy 223

xxxy 223

Suppose we translate this function by

32

3

2


)()()( xxxy 223 3 3 3 2

The same rule works for all functions!


xxxy 223

3

2

xxxy 223

The equation for the translation by is

32


This means that on the graph, x can never be 0

e.g. (a) Sketch the graph of

(b) Write down the translation of the

graph by

(c) Sketch the new graph.

xy 1

2

1

Solution: (a)

• The graph has 180 rotational symmetry about the origin e.g.

• x = 0 ( infinity )0

1y

• As x increases, y decreases

;11 yx

e.g. x = 1 y = 1; 2

1yx = 2 ;

3

1yx = 3

3

13 yx;

2

12 yx


xy

1

xy

1

• As x increases, y decreases

xy 1The graph

of

On this graph, the x-and y-axes form asymptotes

Asymptotes are lines that a graph approaches as x or y approaches infinity.

• Rotational symmetry


xy

1For the graph of

xy

1

•As ,x 0y


,y•As 0x

xy

1

xy

1For the graph of


xy

1

We usually show the asymptotes with a broken line.

0y

0x

The equations of the asymptotes must always be given

For the graph of x

y1


2

1(b) Translating by

gives xy 1 2

11 x

y

The asymptotes have also been translated 10 xx 20 yy

2y

1x


21

1

xySo the graph of is

2y

1x


qpxfy )(

SUMMARY

q

p)(xfy The function given by translating any

function by the vector is given

by

So, to find the translated function, we

replace by andx px

add q

( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )


3xy

Using the same axes for each pair, sketch the following functions:

Exercise

1. 32 xyand

2. x

y1

and 21

x

y

3. xy and 3 xy

Check your answers using “Autograph” or a graphical calculator.


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.


The function is an example of a cubic function.

3xy

3xy

• The graph has 180 rotational symmetry about the origin.


e.g. is a translation of of23 3 )(xy 3xy

3xy 23 3 )(xy

2

3Translation

s

3

2


qpxfy )(

SUMMARY

q

p)(xfy The function given by translating any

function by the vector is given

by

So, to find the translated function, we

replace by andx px

add q

( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )


)()()( xxxy 223 3 3 3 2

The equation for the translation by is

32

The same rule works for all functions!


xxxy 223

xxxy 223

e.g.


xy

1

We usually show the asymptotes with a broken line.

0y

0x

The equations of the asymptotes must always be given

The graph of x

y1


21

1

xyThe graph of is

2y

1x

7: More Graphs and Translations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core...

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