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

2. Module C1
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3. x = 0  y = 0, so the graph goes through the origin.
The function 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. x
As x increases, y increases quickly
e.g. x = 1  y = 1;
x = 2  y = 8
The graph has 180 rotational symmetry about the origin
When sketching a graph, we try not to PLOT points. We want the general shape not an accurate drawing.
e.g. x = -1  y = -1;
x = -2  y = -8

4. We have seen that the quadratic function
is a translation of by
In a similar way, is a translation of

5. is another cubic function
Suppose we translate this function by

6. is another cubic function
The equation for the translation by is
The same rule works for all functions!

7. x = 0  ( infinity ) x = 2  ; x = 3  e.g. (a) Sketch the graph of
(b) Write down the translation of the graph by
(c) Sketch the new graph.
x = 0  ( infinity )
Solution: (a)
This means that on the graph, x can never be 0
As x increases, y decreases
x = 2  ;
x = 3 
e.g. x = 1  y = 1;
The graph has 180 rotational symmetry about the origin e.g.

8. The graph of
As x increases, y decreases
Rotational symmetry
On this graph, the x-and y-axes form asymptotes
Asymptotes are lines that a graph approaches as x or y approaches infinity.

9. For the graph of As

10. For the graph of As

11. For the graph of
We usually show the asymptotes with a broken line.
The equations of the asymptotes must always be given

12. (b) Translating by gives
The asymptotes have also been translated

13. So the graph of is

14. SUMMARY
The function given by translating any function
by the vector is given by
So, to find the translated function, we
replace by and
add
( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )

15. Exercise
Using the same axes for each pair, sketch the following functions: 1.
and
and 2. 3.
and
Check your answers using “Autograph” or a graphical calculator.

17. 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.

18. The function is an example of a cubic function.
The graph has 180 rotational symmetry about the origin.

19. e.g. is a translation of of
Translations

20. SUMMARY
The function given by translating any function
by the vector is given by
So, to find the translated function, we
replace by and
add
( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )

21. The equation for the translation by is
The same rule works for all functions!
is another cubic function
e.g.

22. We usually show the asymptotes with a broken line.
The equations of the asymptotes must always be given
The graph of

23. The graph of is