1. Combining several transformations
The order is very important
This power point will demonstrate which order to do them in
This will then be applied to a Sine Curve

2. Y= x2

3. Y= (2x)2
Horizontal stretch factor ½
HIVO HOVIS
Horizontal In side – Horizontal Opposite

4. Y= (x–3)2
Horizontal translation +3
HIVO HOVIS
Horizontal In side – Horizontal Opposite

5. Y= (2x–3)2
Horizontal translation +3
Horizontal stretch factor ½

6. So the order is:
Translate horizontally left or right
Stretch horizontally
Stretch Vertically
Translate vertically
HIVO
Horizontal Inside Vertical Outside
HOVIS
Horizontal Opposite Vertical Is Same

7. x y sin 2 = e.g. 1 Sketch the graph of the function
Solution: We can use the fact that is a stretch of x y
sin 2 =
is a stretch of s.f. 2, parallel to the y-axis. 90
180
270
360
HIVO Horizontal Inside Vertical Outside
HOVIS Horizontal Opposite Vertical Is Same

8. x y sin 2 = e.g. 1 Sketch the graph of the function
Solution: We can use the fact that is a stretch of x y
sin 2 =
is a stretch of s.f. 2, parallel to the y-axis. 90
180
270
360
The scale factor of the stretch gives the amplitude of the function.

9. e.g. 2 Sketch the graph of the function
Solution:
stretch of s.f. , parallel to the x-axis. So,
is a 90
180
270
360
HIVO Horizontal Inside Vertical Outside
HOVIS Horizontal Opposite Vertical Is Same

10. e.g. 2 Sketch the graph of the function
Solution:
stretch of s.f. , parallel to the x-axis. So,
is a 90
180
270
360
The period of is

11. Exercises
1. Give the equation of the function that is shown on the sketch below.
Ans:
HIVO Horizontal Inside Vertical Outside HOVIS Horizontal Opposite Vertical Is Same

12. Sketch both functions on the same axes for the interval
2. Describe in words the transformation
Solution:
A stretch of s.f. 2 parallel to the x-axis. 90
180
270
360
–90
–180
HIVO Horizontal Inside Vertical Outside HOVIS Horizontal Opposite Vertical Is Same

13. showing the scales clearly. What is the period of the function?
Sketch the graph of for
showing the scales clearly. What is the period of the function?
Solution:
The period is
As it has been stretched horizontally by 
HIVO Horizontal Inside Vertical Outside HOVIS Horizontal Opposite Vertical Is Same

14. Reflection in the x-axis
Every y-value changes sign when we reflect in the x-axis e.g. 90
180
270
360
–90 90
180
270
360
–90
So,
In general, a reflection in the x-axis is given by

15. 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

16. (i) a stretch of s.f. 2 parallel to the x-axis
Solution:
(i) a stretch of s.f. 2 parallel to the x-axis 90
180
270
360 90
180
270
360
stretch

17. Brackets aren’t essential here but they make it clearer.
(ii) a translation of :
Brackets aren’t essential here but they make it clearer. 90
180
270
360
translate

18. (iii) a reflection in the x-axis
(ii) a translation of :
(iii) a reflection in the x-axis x 90
180
270
360 90
180
270
360 x
translate
reflect

19. 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
( Draw for )
(b) y = cosx to y = 2cos(x – 30)

20. Stretch s.f. parallel to the x-axis
Solutions:
(a) to
Stretch s.f. parallel to the x-axis
Translation 90
180
270
360 90
180
270
360

21. (b) to
Solutions:
Translation parallel to the x-axis
Vertical stretch factor 2

22. We need to sketch the graph of y = 3sin(5t+90)
Trig Transformations y
y=sint t
jhvjvjvh
We need to sketch the graph of y = 3sin(5t+90)

23. Trig Transformations y
period = 360
y=sint t
Crosses x axis at 0, 180, 360, 540

24. Trig Transformations y
y=sin(t+90)
Translate horizontally left or right t
Horizontal translation of -90
Inside = horizontal opposite
Crosses x axis at 90, 270, 450

25. Trig Transformations y
Wave frequency = 5
Period =
= 72
y=sin(5t+90) 72 t
Stretch horizontally
Horizontal stretch of 
Inside = horizontal opposite
Crosses x axis at 18, 54, 90

26. Trig Transformations y
Stretch Vertically
y=3sin(5t+90) t
Vertical stretch of factor 3
Outside = vertical same

27. Trig Transformations y
y=3sin(5t+90)+2 t +2
Vertical translation of +2
Outside = vertical same

28. Sketch the graph of
y = 1sin(t + 45)
y = 2sin(t + 30)
y = 3sin(2t – 90)
y = 4sin(3t + 60)

29. y = 1sin(t + 45) y = 2sin(t + 30) 1 2 y = 3sin(2t – 90)
Translate horizontally by –30 Stretch vertically factor of 2
Translate horizontally by –45
y = 3sin(2t – 90)
y = 4sin(3t + 60) 3 4
Translate horizontally by +90 Stretch horizontally by ½ Stretch vertically factor of 3
Translate by horizontally –60 Stretch horizontally by 1/3 Stretch vertically factor of 3

30. Finding the equation from a graph
Y= Asin(f t + ) + c
The mean line is at y = 4
The graph has been translated vertically by +4

31. Finding the equation from a graph
Y= Asin(f t+ ) + 4
The graph has been translated vertically by +4
So c = 4

32. Finding the equation from a graph
Y= Asin(f t+ ) + 4 2
The graph has an amplitude of 2
So the graph has been stretched vertically by a factor of 2

33. Finding the equation from a graph
Y= 2sin(f t+ ) + 4 2
The graph has an amplitude of 2
So A = 2

34. Finding the equation from a graph
Y= 2sin(f t+ ) + 4
180
The period = 180
Frequency f =

35. Finding the equation from a graph
Y= 2sin(2t + ) + 4
180
f =

36. a = 2 x 45 = 90 Finding the equation from a graph Y= 2sin(2t + a) + 4
The start of the sine graph is at t = 45.
But this has been stretched by a factor of .
a = 2 x 45 = 90

37. Finding the equation from a graph
Y= 2sin(2t – 90) + 4
As the sin graph has been translated to the RIGHT
then a = –90

38. 2Sin(x+45) Sin(x+30) 2Sin(x+60)+1 Sin(x-15)+2
Sine graph which has been translated horizontally by -45 and stretched vertically by factor of 2
Sin(x+30)
Sine graph which has been translated horizontally by -30
2Sin(x+60)+1
Sine graph which has been translated horizontally by -60 and stretched vertically by factor of 2 and then translated vertically by +1
Sin(x-15)+2
Sine graph which has been translated horizontally by +15 and translated vertically by +2

39. Sin(x–30)–1 2Sin(x–30) 3Sin(x+45)–1 3Sin(x+90)+ 0.5
Sine graph which has been translated horizontally by +30 and translated vertically by -1
2Sin(x–30)
Sine graph which has been translated horizontally by +30 and stretched vertically by factor of 2
3Sin(x+45)–1
Sine graph which has been translated horizontally by -45 and stretched vertically by factor of 2 and then translated vertically by -1
3Sin(x+90)+ 0.5
Sine graph which has been translated horizontally by +90 and stretched vertically by factor of 3 and then translated vertically by +0.5