1. 1
Oct 29­3:50 PM
Transformations of graphs
To explore geometrical transformations and their
effects on the graphs of functions
To relate transformations with the variations in the
equation of a function.
General objectives:

2. 2
Oct 29­3:50 PM
Transformations of graphs
To explore vertical and horizontal translations and
their effects on the graphs of functions
To relate translations with the variations in the
equation of a function.
TRANSLATIONS
Lesson 1

3. 3
Oct 29­3:50 PM
Based on the graph of f(x) , draw the graph of f(x)+3

4. 4
Oct 29­3:50 PM
What will the graph of f(x) ­ 2 be ?
Vertical translation.ggb

5. 5
Oct 29­3:50 PM
Based on the graph of f(x) , draw the graph of f(x) ­ 1
y=f(x)

6. 6
Oct 29­3:50 PM
This is the graph of y = f(x) .Sketch the graph of
f(x) ­ 4 y=f(x)

7. 7
Oct 29­3:50 PM
Conclusions: If y = f (x)
y = f (x ) + c translates vertically the graph of
y = f (x), c units.
• If it moves upwards.
• If it moves downwards.
c > 0
c < 0
translation vector:

8. 8
Oct 29­3:50 PM
Based on the graph of , draw the graph of y = f (x ­ 2)
f(x­2)=
y=f(x)
If , find f (x ­ 2)

9. 9
Oct 29­3:50 PM
Based on the graph of , draw the graph of y=f(x­2)
f(x­2)=
y=f(x)
Horizontal translation.ggb

10. 10
Oct 29­3:50 PM
y=f(x)
Use your GDC to draw the graph of
and

11. 11
Oct 29­3:50 PM
y=f(x)
andCompare
What can you predict about the graph of g?

12. 12
Oct 29­3:50 PM
asymptotes?

13. 13
Oct 29­3:50 PM
Conclusions:
y= f( x ­ b)
y= f( x + b)
y= f( x )If
b >0
translates horizontally the graph of
y= f( x ), b units to the right.
translates horizontally the graph of
y= f( x ), b units to the left.
translation vector:
translation vector:

14. 14
Oct 29­3:50 PM
Exercise 1: Sketch the graph of: f(x) ­ 2 ; f (x ­ 3)
f(x)

15. 15
Oct 29­3:50 PM
Exercise 2: Sketch the graph of the function
Hence, sketch the graph of

16. 16
Oct 29­3:50 PM
Exercise 3: • Sketch the graph of
• Sketch the graph of • Find g(x) in its simplest form.

17. 17
Oct 29­3:50 PM
Exercise 4:
• Sketch the graph of and indicate clearly any asymptote.
• Sketch the graph of and indicate clearly any asymptote.

18. 18
Oct 29­3:50 PM
Exercise 5:
• Sketch the graph of and indicate clearly any asymptote.
• Sketch the graph of and indicate clearly any asymptote.

19. 19
Oct 29­3:50 PM
Lesson 2
Given sketch the graphs of the following functions on the
same set of axes.

20. 20
Oct 29­3:50 PM
Transformations of graphs
To explore stretches and reflections and their
effects on the graphs of functions
To relate stretches and reflections with the
variations in the equation of a function.
STRETCH
Lesson 2
REFLECTIONS

21. 21
Oct 29­3:50 PM
What is the effect on the graph of that
will produce if
y= f(x)
y= a f(x) ?
"a"
Set up your GDC in "degrees".
Prepare domain :
With your calculator , plot the graphs of

22. 22
Oct 29­3:50 PM
y = sin x
Vertical stretch.ggb

23. 23
Oct 29­3:50 PM
Verify your conclusion for the graphs of

24. 24
Oct 29­3:50 PM
Conclusions:
y= a f( x )
y= f( x )If
a >1
stretches vertically the graph of
y= f( x ), scale factor:
stretches vertically the graph of
y= f( x ), scale factor:
a
y= f( x )
1
a
1
a

25. 25
Oct 29­3:50 PM
Use your calculator to draw the graph of , for
On the same grid draw
What effect produces the "2" of f(2x) on the graph of f(x)?

26. 26
Oct 29­3:50 PM
y= sin (2x) horizontal stretch , scale factor
y= sin x y= sin (2x)
Now , draw and
Horizontal stretch.ggb

27. 27
Oct 29­3:50 PM
y= sin (½x) horizontal stretch , scale factor 2
y= sin x y= sin (½x)

28. 28
Oct 29­3:50 PM
Conclusions: y= f( x )If
a >1
y= f( ax ) stretches horizontally the graph of
y= f( x ), scale factor: 1
a
y= f( x )1
a
stretches horizontally the graph of
y= f( x ), scale factor: a

29. 29
Oct 29­3:50 PM
Exercise 1:
• The function f is defined by
• Sketch the function with the help of your GDC.
• Describe the geometric transformation that
will apply to the graph of f.

30. 30
Oct 29­3:50 PM
Use your calculator to draw the graphs of and
What geometrical transformation does -f(x) represent?

31. 31
Oct 29­3:50 PM
Based on the graph of y= f(x) , draw the graph of y = ­ f(x).

32. 32
Oct 29­3:50 PM
Write down the equations of both lines .

33. 33
Oct 29­3:50 PM
Reflect the following function about the y-axis.

34. 34
Oct 29­3:50 PM
Complete:
g(2) = f (.....) g(1) = f (.....) g(­1) = f (.....)
g(­2) = f (.....) g(x) = f (.......)
y = f(x)y =g(x)

35. 35
Oct 29­3:50 PM
Conclusions:
y = ­f( x )
y= f(­ x )
y= f( x )If
reflects the graph of y= f( x ),
about the x-axis.
reflects the graph of y= f( x ),
about the y-axis.

36. 36
Oct 29­3:50 PM
f(x)
f(x­2)
f(x)+3
2f(x)
Match graphs and formulae:

37. 37
Oct 29­3:50 PM
f(x)
f(x)+a
b f(x)+c b f(x)
f(x+a)

38. 38
Oct 29­3:50 PM
y = f(x)
y = a f(x)
y = f(ax)

39. 39
Oct 29­3:50 PM
Translations
vertical translation
horizontal translation
Stretchs
vertical stretch (a)
horizontal stretch ( )
Reflections
reflection x-axis
reflection y-axis
c
ALibraryOfFunctionsWithTransformations.nbp

40. 40
Oct 29­3:50 PM
To revise this topic at home:
http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@trfns&lid=136
http://archives.math.utk.edu/visual.calculus/0/shifting.7/index.html
When you feel ready, self-assessment:

41. Attachments
Vertical translation exponential.ggb
Horizontal translation quadratic.ggb
ALibraryOfFunctionsWithTransformations.nbp
Horizontal stretch.ggb
Vertical translation.ggb
Horizontal translation.ggb
Vertical stretch.ggb