Quadratic Equations and Functions

1. Quadratic Equations and Functions The ones with the little two above and to the right of the x

2. Standard Form ax2 + bx + c Constant term Quadratic term Linear term

3. Put it in Standard Form • (2x + 3) (x – 4) • x2 + 5x -2x2 + 7 – 3x • (x – 5) ( 3x -1) • x(2x + 4) Now Identify the Quadratic, Linear and Constant Terms

4. Building Quadratic Functions • You can build a quadratic function by multiplying 2 linear functions g(x) = 3x + 4 f(x) = 2x + 9 f(x) g(x) =(2x+9)(3x+4) f(x) g(x) = (6x2 +35x +36)

5. Parent Quadratic f(x) = x2 Parabola Axis of Symmetry Vertex

6. Find the Axis of Symmetry x = -1

7. Find the Vertex The Vertex always lies on the axis of symmetry x = -1 (-1,-5)

8. You can find a quadratic model using 3 points Standard form is ax2 + bx + c = y If I know 3 sets of coordinates (x ,y) Then I can substitute to get 3 equations with 3 (a,b,c) unknowns And solve for (a,b,c)

9. Example a( x)2 + b(x) + c = y a( 2)2 + b(2) + c = 3 a( 3)2 + b(3) + c = 13 a( 4)2 + b(4) + c = 29

10. Example a(4) + b(2) + c = 3 a(9) + b(3) + c = 13 a(16)+ b(4) + c = 29

11. Example Let’s get rid of c ! a(4) + b(2) + c = 3 a(9) + b(3) + c = 13 a(16)+ b(4) + c = 29

12. Let’s move the coefficients out front 4a + 2b + c = 3 a(4) + b(2) + c = 3 9a + 3b+ c = 13 a(9) + b(3) + c = 13 16a + 4b + c = 29 a(16)+ b(4) + c = 29

13. Pair em’ up -1( ) 4a + 2b + c = 3 -1( ) ( ) 4a + 2b + c = 3 9a + 3b+ c = 13 16a + 4b + c = 29

14. Add em -4a + -2b + -c = -3 -4a + -2b + -c = -3 9a + 3b + c = 13 16a + 4b + c = 29 12a + 2b = 26 5a + b = 10

15. Lets get rid of b 12a + 2b = 26 -2( ) 5a + b = 10

16. Solve for a 12a + 2b = 26 -10a +-2b = -20 2a = 6 a = 3

17. Solve for b a = 3 -10(3) +(-2b) = -20 -30 + (-2b) = -20 (-2b) = 10 b = -5

18. Solve for c b = -5 a = 3 4a + 2b + c = 3 4(3) + 2(-5) + c = 3 12 + -10 + c = 3 2 + c = 3 c = 1

19. Write the quadratic model c = 1 b = -5 a = 3 3( x)2 + -5(x) + 1 = y

20. Do Now! Page 237 Problems 1 – 15 Problems 16,20,24,26

21. Parabolas Pair-a-bowl-ahs

22. Parent Quadratic f(x) = x2 Parabola Axis of Symmetry Vertex

23. y = x2and y = ½ x2

24. y = x2and y = - x2

25. y = x2 , y = x2 +2x, y = x2 +4x and y = x2 +6x

26. x = 0 x = -1x = -2 X = -4 y = x2y = x2 +2x y = x2 +4x y = x2 +6x

27. x = 0 x = -1/2x = -1 X = -3/2 y = 2x2y = 2x2 +2x y = 2x2 +4x y = 2x2 +6x

28. Vertex of a Parabola

29. Compare Quadratic Absolute value

30. Graph a Quadratic

31. Graph a Quadratic • Evaluate the function at another point • Graph that point and it’s reflection across the axis of symmetry • Sketch the curve

32. Example Graph -1 = x2+2x-y y = x2+2x+1 a = 1 b = 2 c =1 y = (-1)2+2(-1)+1 y = 1+-2+1 = 0 Vertex = (-1 , 0)

33. Example

34. Example Graph Vertex = (-1 , 0) y -intercept = c y -intercept =1 Graph (0 , 1) Axis of Symmetry x = -1 Vertex is one unit to the right Reflection is one unit to the left Graph (-2 , 1)

35. Example

36. Example Graph evaluate at x = 1 y = (1)2+2(1)+1 y = 1 + 2 + 1 = 4 Graph (1 , 4) Axis of Symmetry x = -1 Vertex is two units to the right Reflection is two units to the left Graph (-3 , 4)

37. Trace the curve

38. Example

39. Do Now! • Page 244 • Problems 1- 4, 7, 10 – 19, 22, 24, 37-39