1. 9.1: QUADRATIC GRAPHS: Quadratic Function: A function that can be written in the form y = ax 2 +bx+c where a ≠ 0. Standard Form of a Quadratic: A function written in descending degree order, that is ax 2 +bx+c.

2. Quadratic Parent Graph: The simplest quadratic function f(x) = x 2. Parabola: The graph of the function f(x) = x 2. Axis of Symmetry: The line that divide the parabola into two identical halves Vertex: The highest or lowest point of the parabola.

3. Minimum: The lowest point of the parabola. Maximum: The highest point of the parabola. Line of Symmetry Vertex = Minimum

4. GOAL:

5. IDENTIFYING THE VERTEX: The vertex will always be the lowest or the highest point of the parabola. Ex: What are the coordinates of the vertex? 1) 2)

6. SOLUTION: The vertex will always be the lowest or the highest point of the parabola. Vertex: ( 0, 3) x =0 Line of Symmetry, y =3 is the Maximum

7. SOLUTION: The vertex will always be the lowest or the highest point of the parabola. Vertex: ( -2, -3) x = -2 Line of Symmetry, y = -3 is the Minimum

8. GRAPHING y = ax 2 : Remember that when we do not know what something looks like, we always go back to our tables.

9. GRAPHING: Xy = (1/3)x 2 y -2 (1/3)∙(-2) 2 (1/3)∙(-1) 2 0 (1/3)∙(0) 2 0 = 0 1 (1/3)∙(1) 2 2 (1/3)∙(2) 2

10. GRAPHING: Xy -2 0 1 2 Domain (-∞, ∞) 0 Range: (0, ∞)

11. USING TECHNOLOGY:

12. y = x 2 Graphing calculators can aid us on looking at properties of functions: Vertex: (0,0) Domain: (- ∞, ∞) Range: (0, ∞)

13. USING TECHNOLOGY: y = 4x 2 Graphing calculators can aid us on looking at properties of functions: Vertex: (0,0) Domain: (- ∞, ∞) Range: (0, ∞)

14. USING TECHNOLOGY: y = -4x 2 Graphing calculators can aid us on looking at properties of functions: Vertex: (0,0) Domain: (- ∞, ∞) Range: (- ∞, 0)

15. USING TECHNOLOGY: Graphing calculators can aid us on looking at properties of functions: Vertex: (0,0) Domain: (- ∞, ∞) Range: (0, ∞)

16. USING TECHNOLOGY: Graphing calculators can aid us on looking at properties of functions: Vertex: (0,0) Domain: (- ∞, ∞) Range: (- ∞, 0)

17. y=x 2 y=4x 2 y= -4x 2 Notice: if coefficient is positive: Parabola faces UP if coefficient is Negative: Parabola faces DOWN if coefficient is > 1: Parabola is Skinny if coefficient is Between 0 and 1: Parabola is wide

18. USING TECHNOLOGY: What is the difference and Similarities of : 1) y = 4x 2 +22) y = 4x 2 -2 3) y = -4x 2 +24) y = -4x 2 -2

19. y = 4x 2 +2 y = 4x 2 -2 Notice: Y = a(x-h) 2 +k +k shift up Y = a(x-h) 2 -k -k shift down +a faces up

20. y = -4x 2 +2 y = -4x 2 -2 Notice: Y = a(x-h) 2 +k Y = -a (x-h) 2 +k +k shift up Y = -a(x-h) 2 -k -a faces down -k shift down

21. REAL-WORLD: A person walking across a bridge accidentally drops and orange into the rives below from a height of 40 ft. The function h = -16t 2 + 40 gives the orange’s height above the water, in feet, after t seconds. Graph the function. In how many seconds will the orange hit the water?

22. GRAPHING: th= -16t 2 +40h 0 -16∙(0) 2 +40 =40 40 -16∙(1) 2 +40 =24 24 1 2 -16∙(2) 2 +40 -24 = -24 Notice: We stop after we get a negative height as we Cannot go beyond the ground.

23. SOLUTION: Once again: Seconds (t) must start at 0  t = 0 Height (h) must stop at 0  h = 0 Thus: our orange will take about 1.6 seconds to hit the ground. Seconds (t) Height (h)

24. VIDEOS: Quadratic Graphs and Their Properties Interpreting Quadratics: http://www.khanacademy.org/math/algebra/quadratics/q uadratic_odds_ends/v/algebra-ii--shifting-quadratic- graphs http://www.khanacademy.org/math/algebra/quadratics/ graphing_quadratics/v/graphing-a-quadratic-function Graphing Quadratics:

25. VIDEOS: Quadratic Graphs and Their Properties Graphing Quadratics: http://www.khanacademy.org/math/algebra/quadratics/gr aphing_quadratics/v/quadratic-functions-1

26. CLASSWORK: Page 537-539: Problems: 1, 2, 3, 4, 7, 8, 10, 13, 19, 27, 28, 34  39.