The document discusses how shifting curves vertically and horizontally affects their shape and properties. Vertically shifting a curve by a factor of k, where k>1 makes the curve steeper and k<1 makes it shallower. Horizontally shifting by a factor of k also affects the steepness in the same way and changes the domain. Examples of shifting simple curves like y=x are shown to illustrate these effects on the curve shape, intercepts and asymptotes.

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