The document discusses odd and even functions. It defines odd functions as those where f(-x) = -f(x), and provides examples such as y = x and y = x^3 + x^7. It proves that y = x^3 + x^7 is an odd function. Even functions are defined as those where f(-x) = f(x), with examples like y = x^2 + 4. It proves y = x^2 + 4 is an even function by showing f(-x) = f(x).

1. Symmetry

2. Odd Functions
Symmetry
f x   f  x

3. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 

4. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x

5. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”

6. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function

7. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7

8. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7 f x  x  x
3 7

9. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7 f x  x  x
3 7
  x3  x7

10. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7 f x  x  x
3 7
  x3  x7
   x3  x7 

11. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7 f x  x  x
3 7
  x3  x7
   x3  x7 
  f  x

12. Odd Functions
Symmetry
f x   f  x
The curve has point symmetry about the origin
 If you spin it 180 it looks the same 
1
e.g. y  x , y  , y  x 7  x 5 , y  3 x 9  2 x 7
3
x
Note: “all the powers are odd”
e.g. Prove that y  x 3  x 7 is an odd function
f  x   x3  x 7 f x  x  x
3 7
  x3  x7
   x3  x7 
  f  x  odd function

13. Even Functions
f x  f  x

14. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 

15. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2

16. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”

17. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function

18. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4

19. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4 f x  x  4
2

20. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4 f x  x  4
2
 x2  4

21. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4 f x  x  4
2
 x2  4
 f  x

22. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4 f x  x  4
2
 x2  4
 f  x  even function

23. Even Functions
f x  f  x
The curve has line symmetry about the y axis
 the y axis is an axis of symmetry 
e.g. y  x 2 , y  x 2  4, y  3 x 6  2 x 4  27 x 2
Note: “all the powers are even”
e.g. Prove that y  x 2  4 is an even function
f  x   x2  4 f x  x  4
2
 x2  4
 f  x  even function
Exercise 3C; 1aceg, 2, 4aceg, 5, 6bdfh, 8adf, 9, 10*