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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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 x3 , y , y x 7 x 5 , y 3 x 9 2 x 7
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
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*