9 1 QUADRATIC GRAPHS Quadratic Function A function

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.

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.

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

GOAL:

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)

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

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

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

GRAPHING: X -2 y = (1/3)x 2 (1/3)∙(-2)2 -1 (1/3)∙(-1)2 0 (1/3)∙(0)2 = 0 1 (1/3)∙(1)2 2 (1/3)∙(2)2 y 0

GRAPHING: X y -2 -1 0 1 2 Domain (-∞, ∞) Range: (0, ∞) 0

USING TECHNOLOGY:

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

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

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

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

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

y=4 x 2 y=x 2 y= -4 x 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

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

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

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

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 = -16 t 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?

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

SOLUTION: Once again: Height (h) 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)

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

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

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