“Teach A Level Maths”
Vol. 1: AS Core Modules
7: More Graphs and
Translations
© Christine Crisp
More Graphs and Translations
Module C1
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More Graphs and Translations
 The function y  x 3 is an example of a cubic function.
To sketch the graph we notice the following:
•
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
sketching
graph,
we
• When
The graph
has a
180
rotational
try
symmetry
not to PLOT
aboutpoints.
the origin
Wexwant
e.g.
= -1 the
 ygeneral
= -1; shape
not an accurate drawing.
x = -2  y = -8
x
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We have seen that the quadratic function
y  (x  3)  2 is a translation of y  x
2
2
 3
by  
 2
 In a similar way, y  (x  3) 3  2 is a translation
of y  x 3
2
3
yx
3
y  (x  3) 3  2
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y  x 3  x 2  2 x is another cubic function
y  x3  x2  2x
2
3
Suppose we translate this function by  3
 2
More Graphs and Translations
y  x 3  x 2  2 x is another cubic function
y  x3  x2  2x
2
3
The equation for the translation by  3 is
 2
y  ( x  3)  ( x  3)  2( x  3)  2
3
2
The same rule works for all functions!
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1
e.g. (a) Sketch the graph of y 
x
  1
(b) Write down the translation of the graph by   2
 
(c) Sketch the new graph.
1
Solution: (a) • x = 0  y   
0
•
•
( infinity )
This means that on the graph, x can never be 0
As x increases, y decreases
1
1
e.g. x = 1  y = 1; x = 2  y  ; x = 3  y 
2
3
The graph has 180 rotational symmetry about
the origin e.g.
1
1
x  1  y  1; x  2  y   ; x  3  y  
2
3
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The graph of y  1
x
•
As x increases,
y decreases
•
Rotational
symmetry
y
1
x
On this graph, the
x-and y-axes form
asymptotes
Asymptotes are lines that a graph approaches
as x or y approaches infinity.
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For the graph of y 
• As x    , y  0
1
x
y
1
x
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For the graph of y 
• As y    , x  0
1
x
y
1
x
More Graphs and Translations
For the graph of y 
1
x
y
We usually show the
asymptotes with a
broken line.
y0
The equations of the
asymptotes must
always be given
x0
1
x
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1
  1
y

(b) Translating
by   gives
x
 2
y  1 2
x 1
The asymptotes have also been translated
x0

x  1
y0

y  2
x  1
y  2
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1
 2 is
So the graph of y 
x 1
x  1
y  2
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SUMMARY
The function given by translating any function
y  f (x) by the vector
 p is given by
q
 
y  f ( x  p)  q
So, to find the translated function, we
 replace
 add
x
by
x  p and
q
( Notice that adding q is the same as replacing
y by y – q. We’ll need this later. )
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Exercise
Using the same axes for each pair, sketch the
following functions:
1.
y  x3
2.
1
y
x
3.
y x
and
y  x  23
and
1
y  2
x
and
y  x3
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 y  x 3 is an example of a cubic function.
y  x3
•
The graph has 180 rotational symmetry
about the origin.
More Graphs and Translations
Translations
e.g. y  (x  3)  2
3
3

is a translation of y  x of
 2
 
3
2
3
y  x3
y  (x  3) 3  2
More Graphs and Translations
SUMMARY
The function given by translating any function
y  f (x) by the vector
 p is given by
q
 
y  f ( x  p)  q
So, to find the translated function, we
 replace
 add
x
by
x  p and
q
( Notice that adding q is the same as replacing
y by y – q. We’ll need this later. )
More Graphs and Translations
3
2
e.g. y  x  x  2 x is another cubic function
y  x3  x2  2x
The equation for the translation by  3 is
 
 2
y  ( x  3)  ( x  3)  2( x  3)  2
3
2
The same rule works for all functions!
More Graphs and Translations
1
The graph of y 
x
y
We usually show the
asymptotes with a
broken line.
The equations of the
asymptotes must
always be given
1
x
y0
x0
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1
 2 is
The graph of y 
x 1
x  1
y  2