1. Standard 9.0 Determine how the graph of a parabola changes as a, b, and c vary in the equation Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions

2. G RAPHING A Q UADRATIC F UNCTION A quadratic function has the standard form y = ax 2 + bx + c where a  0.

3. G RAPHING A Q UADRATIC F UNCTION The graph is “U-shaped” and is called a parabola.

4. G RAPHING A Q UADRATIC F UNCTION The highest or lowest point on the parabola is called the ver tex. What is another word for highest? What is another word for lowest? In your notes, draw and label the maxima of a parabola. CORRECT!! The highest point is called maxima And the lowest point is called the minima?

5. G RAPHING A Q UADRATIC F UNCTION In general, the axis of symmetry for the parabola is the vertical line through the vertex.

6. G RAPHING A Q UADRATIC F UNCTION These are the graphs of y = x 2 and y =  x 2.

7. G RAPHING A Q UADRATIC F UNCTION The y -axis is the axis of symmetry for both graphs.

8. Graphing a Quadratic Function Graph y = 2 x 2 – 8 x + 6 x = – = – = 2 b 2 a2 a – 8 2(2) y = 2(2) 2 – 8 (2) + 6 = – 2 So, the vertex is (2, – 2). (2, – 2) The x -coordinate is: The y -coordinate is: Find and plot the vertex.

9. VERTEX FORM OF A QUADRATIC FUNCTION G RAPHING A Q UADRATIC F UNCTION CHARACTERISTICS OF GRAPH Vertex form: y = a (x – h) 2 + k For this form, the graph opens up if a > 0 and opens down if a < 0. The vertex is (h, k ). Open up “ + a” Open down “ - a”. Regular, narrow, wide. “a” Centered, horizontal shift left, horizontal shift right “h” Centered, vertical shift up, vertical shift down “k”

10. Is h positive or negative? positive move h units to the right negative move h units to the left Is a positive or negative? positive open up negative open down Is k positive or negative? positive move k units up negative move k units down Transforming the Graph of a Quadratic Function y=a(x-h) 2 +k

11. Quadratic Function Graphing a Quadratic Function

12. y = a(x-h) 2 + k Graphing a Quadratic Function

23. Things to Notice Where is the constant term located? Outside of the power of 2 Inside the power of 2

24. Graphing a Quadratic Function

40. Describe the change from blue to red. Graphing a Quadratic Function

45. ParabolaParabola in foci in motion Graphing a Quadratic Function

47. Graph y = – (x + 3) 2 + 4 1 2 S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = 4 1 2 First graph y=x 2

48. Graphing a Quadratic Function Graph y = – (x + 3) 2 + 4 1 2 S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = 4 1 2 First graph y=x 2

49. Graphing a Quadratic Function Graph y = – (x + 3) 2 + 4 1 2 S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = 4 1 2 First graph y=x 2

50. Graphing a Quadratic Function Graph y = – (x + 3) 2 + 4 1 2 S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = 4 1 2 First graph y=x 2

51. Graphing a Quadratic Function Graph y = – (x + 3) 2 + 4 1 2 S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = 4 1 2 First graph y=x 2

52. Graphing a Quadratic Function Graph y = – (x + 3) 2 + 4 1 2

53. Graphing a Quadratic Function Write an equation that could describe the red Parabola

54. Graphing a Quadratic Function Write an equation that could describe the red Parabola

55. Graphing a Quadratic Function Write an equation that could describe the red Parabola

56. Graphing a Quadratic Function Write an equation that could describe the red Parabola

57. Graphing a Quadratic Function Write an equation that could describe the red Parabola

58. Graphing a Quadratic Function Write an equation that could describe the red Parabola

59. Graphing a Quadratic Function Write an equation that could describe the red Parabola

60. Graphing a Quadratic Function Write an equation that could describe the red Parabola

61. Graphing a Quadratic Function Write an equation that could describe the red Parabola

62. Graphing a Quadratic Function Write an equation that could describe the red Parabola

63. Graphing a Quadratic Function What can you conclude about the constants inside the parenthesis and outside the parenthesis?