1. Transformations

2. Horizontally or Vertically or both
Translations applied to a graph mean move movements
Horizontally or Vertically or both
They can be described using a vector as shown below
Means translate(move) a graph 5 to the right
And translate (move it) 2 up

3. The graph of forms a curve called a parabola
This point
is called the vertex

4. Adding a constant translates up the y-axis
e.g.
The vertex is now ( 0, 3)
has added 3 to the y-values

5. Adding 3 to x gives
We get
Adding 3 to x moves the curve 3 to the left.

6. We can write this in vector form as: translation
Translating in both directions
e.g.
We can write this in vector form as:
translation

7. SUMMARY
The curve
is a translation of by
The vertex is given by

8. SUMMARY
Memory Aid
HIVO – Horizontal Inside the Bracket
Vertical Outside the bracket
HOVIS – Horizontal Opposite of what it says
Vertical Is Same as what it says

9. Exercises: Sketch the following translations of1. 2. 3.

10. 4 Sketch the curve found by translating
by What is its equation?
5 Sketch the curve found by translating
by What is its equation?

11. e.g. The translation of the function by the vector gives the function .
The graph becomes
Means translate horizontally by +2.
So put –2 in the bracket.
HIvo
HOvis

12. e.g. The translation of the function by the vector gives the function .
The graph becomes
Means translate vertically by +1.
So put +1 outside the bracket.
HIVO
HOVIS

13. Vertical Stretches
e.g.1 Consider the following functions:
and
For
For
In transforming from to the y-value has been multiplied by 4

14. e.g.1 Consider the following functions:
and
For
For
In transforming from to the y-value has been multiplied by 4
Similarly, for every value of x, the y-value on
is 4 times the y-value on
is a stretch of scale factor 4 parallel to the y-axis

15. e.g.1 Consider the following functions:
and
HIVO – Horizontal Inside the Bracket
Vertical Outside the bracket
HOVIS – Horizontal Opposite of what it says
Vertical Is Same as what it says
As the 4 is Outside the stretch is applied Vertically

16. The graphs of the functions are as follows:
is a stretch of
by scale factor 4, parallel to the y-axis

17. is a transformation of given by
a stretch of scale factor 4 parallel to the y-axis

18. 3 1 Horizontal stretches Now, for and for
The x-value must be divided by 3 to give the same value of y.
So the x coordinate is stretched by a factor of  parallel to the x axis

19. e.g.1 Consider the following functions:
and
HIVO – Horizontal Inside the Bracket
Vertical Outside the bracket
HOVIS – Horizontal Opposite of what it says
Vertical Is Same as what it says
As the 3 is Inside the stretch is applied Horizontally and is the Opposite. So instead of being 3 times as wide it is a  as wide.

20. SUMMARY
The transformation of to
is a stretch of scale factor 4 parallel to the y-axis or
is a stretch of scale factor
parallel to the x-axis

21. SUMMARY
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.

22. e.g. 2 Describe the transformation of that gives .
Using the same axes, sketch both functions.
Solution: can be written as
so it is a stretch of s.f. 3, parallel to the y-axis
We always stretch from an axis.

23. Exercises
1. (a) Describe a transformation of that gives
(b) Sketch the graphs of both functions to illustrate your answer.
Solution:
(a) A stretch of s.f. 9 parallel to the y-axis.
(b)

24. Transforming the exponential graph y = ex-1 -2 -3 -4 -5 1 2 3 4 5

25. Horizontal stretch scale factor  All the x coordinates are  as wide
y = ex
y = e(2x) -1 -2 -3 -4 -5 1 2 3 4 5 -1 -2 -3 -4 -5 1 2 3 4 5
Horizontal stretch scale factor 
All the x coordinates are  as wide

26. Horizontal stretch scale factor  Vertical stretch scale factor 2
y = ex
y = 2e(2x) -1 -2 -3 -4 -5 1 2 3 4 5
Horizontal stretch scale factor 
Vertical stretch scale factor 2
All the y coordinates are 2x as high

27. y = ex
y = 2e(2x)–4
Horizontal stretch scale factor 
Vertical stretch scale factor 2
Vertical translation

28. Two more Transformations
Reflection in the x-axis
Every y-value changes sign when we reflect in the x-axis e.g.
y=x2
y=–(x)2
So,
In general, a reflection in the x-axis is given by

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

30. 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