The document discusses how shifting curves vertically and horizontally by multiplying or dividing the independent and dependent variables by constants. Vertically shifting a curve by multiplying y by a constant k stretches the curve vertically, leaving the domain unchanged but altering the range. Horizontally shifting by multiplying x by k stretches the curve horizontally, altering the domain but leaving the range unchanged. Examples of shifting simple curves like lines and parabolas are shown to illustrate these transformations.

1. Shifting Curves II

2. Shifting Curves II
y  kf  x  k  1, curve is steeper
OR 0  k  1, curve is shallower
y (curve is stretched vertically)
 f  x
k

3. Shifting Curves II
y  kf  x  k  1, curve is steeper
OR 0  k  1, curve is shallower
y (curve is stretched vertically)
 f  x
k domain unchanged, range altered

4. Shifting Curves II
y  kf  x  k  1, curve is steeper
OR 0  k  1, curve is shallower
y (curve is stretched vertically)
 f  x
k domain unchanged, range altered
y  f  kx  k  1, curve is steeper
0  k  1, curve is shallower
(curve is stretched horizontally)

5. Shifting Curves II
y  kf  x  k  1, curve is steeper
OR 0  k  1, curve is shallower
y (curve is stretched vertically)
 f  x
k domain unchanged, range altered
y  f  kx  k  1, curve is steeper
0  k  1, curve is shallower
(curve is stretched horizontally)
domain altered, range unchanged

6. Shifting Curves II
y  kf  x  k  1, curve is steeper
OR 0  k  1, curve is shallower
y (curve is stretched vertically)
 f  x
k domain unchanged, range altered
y  f  kx  k  1, curve is steeper
0  k  1, curve is shallower
(curve is stretched horizontally)
domain altered, range unchanged
1
y x intercepts  asymptotes
f  x
asymptotes  x intercepts
y 1  y 1
y 1  y 1

7. e.g.  i  on one graph draw
1
a ) y  x, y  x, y  2 x
2
yx
y
x

8. e.g.  i  on one graph draw
1
a ) y  x, y  x, y  2 x
2
yx
y
1
y x
2
x

9. e.g.  i  on one graph draw
1
a ) y  x, y  x, y  2 x
2 y  2x
yx
y
1
y x
2
x

10. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y y  2 x2
y
1
y x
2
x x

11. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y
y
1
y x 1 2
y x
2 2
x x

12. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y y  2 x2
y
1
y x 1 2
y x
2 2
x x

13. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y y  2 x2
y
1
y x 1 2
y x
2 2
x x
y2
(ii ) Sketch x 2   1
4

14. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y y  2 x2
y
1
y x 1 2
y x
2 2
x x
y2
(ii ) Sketch x 2   1 y
4
1
1. basic curve : x  y  1
2 2
–1 1 x
–1

15. e.g.  i  on one graph draw
1 1
a ) y  x, y  x, y  2 x b) y  x 2 , y  x 2 , y  2 x 2
2 y  2x 2
yx y  x2 y y  2 x2
y
1
y x 1 2
y x
2 2
x x
y2
(ii ) Sketch x 2   1 y
4
1
1. basic curve : x  y  1
2 2
2 –1 1 x
y2  y 
2.    , k  2
4 2 –1
stretch vertically by a factor of 2

16. 1
 iii  y 
2  x2

17. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
x

18. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
x

19. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
3. shift up 2 units
2
x

20. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
2
3. shift up 2 units
4. x intercepts become asymptotes
 2
x
2

21. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
2
3. shift up 2 units
4. x intercepts become asymptotes
1
5. y >1 become y <1  2 2
x
2

22. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
2
3. shift up 2 units
4. x intercepts become asymptotes
1
5. y >1 become y <1  2 2
x
6. y <1 become y >1 2

23. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
2
3. shift up 2 units
4. x intercepts become asymptotes
1
5. y >1 become y <1  2 2
x
6. y <1 become y >1 2

24. 1
 iii  y 
2  x2
1. basic curve : y  x 2 y
2. reflect in x axis
2
3. shift up 2 units
4. x intercepts become asymptotes
1
5. y >1 become y <1  2 2
x
6. y <1 become y >1 2

25. 1
 iv  y  x 
x

26. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
x

27. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain

28. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain
- points of intersection

29. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain
- points of intersection
- as many other points as you
need to work out the shape

30. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain
- points of intersection
- as many other points as you
need to work out the shape

31. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain
- points of intersection
- as many other points as you
need to work out the shape

32. 1
 iv  y  x  y
x
1. draw the basic curves :
1
y  x and y 
x
2. add the y values together
* choose key points first x
- x intercepts
note: vertical asymptotes remain
- points of intersection
- as many other points as you
need to work out the shape
Exercise 2J: 1, 2a, 3b, 4c, 5ac, 6b, 7ac, 8, 9