1. “Teach A Level Maths” Vol. 2: A2 Core Modules
15: More Transformations
© Christine Crisp

2. Module C3
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3. We’ll start with a reminder of some examples we’ve already met.
The translations and stretches that we met in AS can be applied to any functions.
In this presentation we will look particularly at the effect on the trig, exponential and log functions of combining transformations.
Try not to use a calculator when doing this topic. Graphs copied from graphical calculators look peculiar unless the scales are chosen very carefully.
If you do use a calculator remember to mark coordinates of all significant points and clearly show the behaviour of the curves near the axes.
We’ll start with a reminder of some examples we’ve already met.

4. e.g. 1 The translation of the function by the vector gives the function
The graph becomes

5. e.g. 2 Describe the transformation of that gives .
Solution: can be written as
so it is a stretch of s.f. 3, parallel to the y-axis

6. e.g. 3 Sketch the graph of the function
Solution:
is a
stretch of s.f. , parallel to the x-axis. So,

7. e.g. 3 Sketch the graph of the function
Solution:
is a
stretch of s.f. , parallel to the x-axis. So,

8. General Translations and Stretches
The function is a translation of by
Stretches
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.

9. Two more Transformations
Reflection in the x-axis
Every y-value changes sign when we reflect in the x-axis e.g. x x
So,
In general, a reflection in the x-axis is given by

10. Reflection in the y-axis
Every x-value changes sign when we reflect in the y-axis e.g. x x
So,
In general, a reflection in the y-axis is given by

11. SUMMARY
Reflections in the axes
Reflecting in the x-axis changes the sign of y
Reflecting in the y-axis changes the sign of x
The examples that follow illustrate combinations of the transformations: translations, stretches and reflections.

12. x has been replaced by 2x:
Combined Transformations
e.g. 1 Describe the transformations of that give the function Hence sketch the function.
Solution:
x has been replaced by 2x:
so we have a stretch of s.f.
parallel to the
x-axis
1 has then been added:
so we have a translation of

13. We do the sketch in 2 stages:
The point on the y-axis . . .
doesn’t move with a stretch parallel to the x-axis

14. We do the sketch in 2 stages:

15. e.g. 2 Describe the transformations of that give
(a) (b)
Solution:
(a) We have
but for (b),
(a) is
a stretch of s.f.

16. y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
a stretch of s.f.
parallel to the
y-axis
a translation of

17. y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
(b) is
a stretch of s.f.
a translation of
parallel to the
y-axis
a translation of

18. y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
(b) is
a stretch of s.f.
a translation of
parallel to the
y-axis
a translation of
a stretch of s.f.

19. y-axis y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
(b) is
a stretch of s.f.
a translation of
parallel to the
y-axis
a translation of
a stretch of s.f.
parallel to the
y-axis

20. y-axis y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
(b) is
a stretch of s.f.
a translation of
parallel to the
y-axis
a translation of
a stretch of s.f.
parallel to the
y-axis

21. The graphs of the functions are:
stretch
translate
(b)
translate
stretch

22. e.g.3 Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).
(i) a stretch of s.f. 2 parallel to the x-axis
then (ii) a translation of
then (iii) a reflection in the x-axis

23. Solution:
(i) a stretch of s.f. 2 parallel to the x-axis
stretch

24. Brackets aren’t essential here but I think they make it clearer.
(ii) a translation of :
Brackets aren’t essential here but I think they make it clearer.
translate

25. (iii) a reflection in the x-axis
(ii) a translation of :
(iii) a reflection in the x-axis x x
translate
reflect

26. Exercises
1. Describe the transformations that map the graphs of the 1st of each function given below onto the 2nd. Sketch the graphs at each stage.
(a) to
(b) to
(c) to
( Draw for )

27. (a) to
Solutions:
( The order doesn’t matter )
Reflection in the y-axis
Stretch s.f. 2 parallel to the y-axis

28. (b) to
Solutions:
( Again the order doesn’t matter )
Translation
Stretch s.f. 2 parallel to the y-axis

29. Solutions:
(c) to
Again the order doesn’t matter.
Stretch s.f. parallel to the x-axis
Translation

30. If a stretch and a translation are in the same direction we have to be very careful.
e.g. A stretch s.f. parallel to the y-axis on
followed by a translation of gives
With the translation first, we get

31. An important example involving stretches is the transformation of a circle into an ellipse.
e.g. Find the equation of the ellipse given by transforming the circle by
(i) a stretch of scale factor 4 parallel to the x-axis, and
(ii) a stretch of scale factor 2 parallel to the y-axis
Method
Rearranging the equation of the circle to y = gives a clumsy expression so we don’t do it.
This means we must change the way we handle the stretch in the y direction.

32. When we had , we stretched by s.f. 2 parallel to the y-axis by writing
i.e. multiplying the r.h.s. by 2.
We could equally well have divided the l.h.s. by 2, so
So, to find the equation of a curve which is stretched by 2 in the y direction, we can replace y by
We are then treating both stretches in the same way.

33. Returning to the example . . .
e.g. Find the equation of the ellipse given by transforming the circle by
(i) a stretch of scale factor 4 parallel to the x-axis, and
(ii) a stretch of scale factor 2 parallel to the y-axis
Replace x by and replace y by
Solution:

34. The ellipse looks like this . . .
If we want to translate the ellipse we use a similar technique
e.g. to translate by replace x by and
replace y by
So,
The answer is usually left in this form.

35. The graphs look like this:x

36. SUMMARY
When we cannot easily write equations of curves in the form
we can obtain stretches of scale factor k by
Replacing x by and replacing y by
we can obtain a translation of by
Replacing x by
Replacing y by

37. Exercises
1. Find the equation of the curve obtained from with the transformations given.
(i) a stretch of s.f. 3 parallel to the x-axis and
(ii) a stretch of s.f. 5 parallel to the y-axis
(iii) followed by a translation of .
2. Find the equation of the curve obtained from with the transformations given.
2. Find the equation of the curve obtained from with the transformations given.
(i) a stretch of s.f. 2 parallel to the x-axis and
(ii) a stretch of s.f. 5 parallel to the y-axis
(iii) followed by a translation of .

38. Solutions:
1 (i) a stretch of s.f. 3 parallel to the x-axis
(ii) a stretch of s.f. 5 parallel to the y-axis
(iii) followed by a translation of .

39. Solutions:
2 (i) a stretch of s.f. 2 parallel to the x-axis
(ii) a stretch of s.f. 5 parallel to the y-axis
( or )
(iii) followed by a translation of
( or )

40. You may have to deal with a function shown only in a drawing ( with no equation given ).
If you are confident about the earlier work, try this one before you look at my solution.

41. The diagram shows part of the curve with equation .x y
Copy the diagram twice and on each diagram sketch one of the following:
(i) (ii)

42. Solution: x y
(i) x y
(ii)

44. 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.

45. General Translations and Stretches
The function is a translation of by
Translations
Stretches
The function is obtained from
by a stretch of scale factor ( s.f. ) ,
parallel to the x-axis.
The function is obtained from
by a stretch of scale factor ( s.f. ) k,
parallel to the y-axis.

46. SUMMARY
Reflections in the axes
Reflecting in the x-axis changes the sign of y
Reflecting in the y-axis changes the sign of x

47. x has been replaced by 2x:
Combined Transformations
e.g. 1 Describe the transformations of that give the function Hence sketch the function.
Solution:
x has been replaced by 2x:
so we have a stretch of s.f.
1 has then been added:
so we have a translation of
parallel to the
x-axis

48. The point on the y-axis doesn’t move with a stretch parallel to the x-axis.
We do the sketch in 2 stages:

49. y-axis e.g. 2 Describe the transformations of that give (a) (b)
Solution:
(a) We have
but for (b),
(a) is
a stretch of s.f.
parallel to the
y-axis
a translation of
(b) is

50. then (iii) a reflection in the x-axis
(i) a stretch of s.f. 2 parallel to the x-axis
then (ii) a translation of
e.g.3 Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).

51. Solution:
(i) a stretch of s.f. 2 parallel to the x-axis
stretch

52. (iii) a reflection in the x-axis
(ii) a translation of :
translate
reflect x
(iii) a reflection in the x-axis

53. SUMMARY
we can obtain stretches of scale factor k by
When we cannot easily write equations of curves in the form
Replacing x by and by replacing y by
we can obtain a translation of by
Replacing x by
Replacing y by

54. (i) a stretch of scale factor 4 parallel to the x-axis,
(ii) a stretch of scale factor 2 parallel to the y-axis
Solution:
Replace x by and replace y by
e.g. Find the equation of the ellipse given by transforming the circle by
(iii) followed by a translation of
Replace x by and replace y by