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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
"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"
More Graphs and Translations
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
More Graphs and Translations
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
More Graphs and Translations
is another cubic function
xxxy 223
xxxy 223
Suppose we translate this function by
32
3
2
More Graphs and Translations
)()()( xxxy 223 3 3 3 2
The same rule works for all functions!
is another cubic function
xxxy 223
3
2
xxxy 223
The equation for the translation by is
32
More Graphs and Translations
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
More Graphs and Translations
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
More Graphs and Translations
xy
1For the graph of
xy
1
•As ,x 0y
More Graphs and Translations
,y•As 0x
xy
1
xy
1For the graph of
More Graphs and Translations
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
More Graphs and Translations
2
1(b) Translating by
gives xy 1 2
11 x
y
The asymptotes have also been translated 10 xx 20 yy
2y
1x
More Graphs and Translations
21
1
xySo the graph of is
2y
1x
More Graphs and Translations
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. )
More Graphs and Translations
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.
More Graphs and Translations
More Graphs and Translations
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.
More Graphs and Translations
The function is an example of a cubic function.
3xy
3xy
• The graph has 180 rotational symmetry about the origin.
More Graphs and Translations
e.g. is a translation of of23 3 )(xy 3xy
3xy 23 3 )(xy
2
3Translation
s
3
2
More Graphs and Translations
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. )
More Graphs and Translations
)()()( xxxy 223 3 3 3 2
The equation for the translation by is
32
The same rule works for all functions!
is another cubic function
xxxy 223
xxxy 223
e.g.
More Graphs and Translations
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
More Graphs and Translations
21
1
xyThe graph of is
2y
1x