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Nature of Roots

Nature of Roots. Nature of Roots. Quadratic Equation: a x 2 + b x + c = 0 ; a  0 Discriminant =  = b 2 – 4 ac.  > 0 Two unequal real roots  = 0 One double real root (Two equal real roots)  < 0 No real roots Note:   0 Real roots.

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Nature of Roots

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  1. Nature of Roots

  2. Nature of Roots Quadratic Equation: ax2 + bx + c = 0 ; a 0 Discriminant =  = b2 – 4ac  > 0 Two unequal real roots  = 0 One double real root (Two equal real roots)  < 0 No real roots Note:   0 Real roots

  3. Transformation of a graph

  4. Translation The original graph is y = f(x) . Let h, k > 0 .

  5. Translation Examples

  6. Reflection The original graph is y = f(x) .

  7. Reflection Examples

  8. Enlargement and Reduction The original graph is y = f(x) .

  9. Enlargement and Reduction Examples

  10. Trigonometric Functions

  11. 30 2 2 60 1 Trigonometric Functions of Special Angles (I) 1 45 1

  12. 0 (0, 1) cos 1 1 (1, 0) (1, 0) 0 (0, 1) undefined 1 tan 0 0 sin 0 0 undefined 1 Trigonometric Functions of Special Angles (II)

  13. A S T C Trigonometric Functions of General Angles (I) II I III IV

  14. Trigonometric Functions of General Angles (II)

  15. Nets of a cube

  16. Nets of a cube Two nets are identical if one can be obtained from the other from rotation (turn it round) or/and reflection (turn it over). An example of identical nets.

  17. Nets of a cube There are a total of 11 different nets of a cube as shown.

  18. Planes of Reflection

  19. Planes of Reflection of a Cube

  20. Planes of Reflection of a Regular Tetrahedron

  21. Axes of Rotation

  22. order of rotational symmetry = 4 order of rotational symmetry = 3 Axes of Rotation of a Cube order of rotational symmetry = 2

  23. order of rotational symmetry = 2 Axes of Rotation of a Regular Tetrahedron order of rotational symmetry = 3

  24. Compare Slopes of Different Lines

  25. undefined slope m7 > 1 m6 = 1 ( = 45) 0 < m5 < 1 x m4 = 0 1 < m3 < 0 m2 = 1 ( = 135) m1 < 1 m1 < m2 < m3 < m4 < m5 < m6 < m7

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