9.1: QUADRATIC GRAPHS:

Quadratic Function: A function that can be written in the form y = ax2+bx+c where a ≠ 0.

Standard Form of a Quadratic: A function written in descending degree order, that is ax2+bx+c.

Quadratic Parent Graph: The simplest quadratic function f(x) = x2.

Parabola: The graph of the function f(x) = x2.

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 parabola. Line of Symmetry

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 = ax2: Remember that when we do not know what something looks like, we always go back to our tables.Ex: Graph y = x2. Make a table and provide the Domain and Range.

GRAPHING:X y = (1/3)x2 y

-2 (1/3) (-2)∙ 2 = 𝟒𝟑

(1/3) (-1)∙ 2 = 𝟏𝟑-1

0 (1/3) (0)∙ 2 0 = 0

1 (1/3) (1)∙ 2 𝟏𝟑=

2 (1/3) (2)∙ 2 =

GRAPHING: X y-2

-1

0

1

2

Domain (-∞, ∞)

𝟒𝟑𝟏𝟑

0 𝟏𝟑

Range: (0, ∞)

USING TECHNOLOGY: What is the difference and Similarities of :

1) y = x2

2) y = 4x2 3) y = -4x2

4) y = x2 5) y = - x2

USING TECHNOLOGY:

y = x2

Graphing calculators can aid us on looking at properties of functions:

Vertex: (0,0)

Domain: (- ∞, ∞)

Range: (0, ∞)

USING TECHNOLOGY:

y = 4x2

Graphing calculators can aid us on looking at properties of functions:

Vertex: (0,0)

Domain: (- ∞, ∞)

Range: (0, ∞)

USING TECHNOLOGY:

y = -4x2

Graphing calculators can aid us on looking at properties of functions:

Vertex: (0,0)

Domain: (- ∞, ∞)

Range: (- ∞, 0)

USING TECHNOLOGY:

y = x2

Graphing calculators can aid us on looking at properties of functions:

Vertex: (0,0)

Domain: (- ∞, ∞)

Range: (0, ∞)

USING TECHNOLOGY:

y = - x2

Graphing calculators can aid us on looking at properties of functions:

Vertex: (0,0)

Domain: (- ∞, ∞)

Range: (- ∞, 0)

y=x2 y=4x2

y= -4x2

y= x2

y = - x2

Notice:if coefficient is positive: Parabola faces UP if coefficient is Negative: Parabola faces DOWNif coefficient is > 1: Parabola is Skinnyif coefficient is Between 0 and 1: Parabola is wide

USING TECHNOLOGY: What is the difference and Similarities of :

1) y = 4x2+2 2) y = 4x2-2

3) y = -4x2+2 4) y = -4x2-2

y = 4x2+2

y = 4x2-2

Notice: Y = a(x-h)2 +k

Y = a(x-h) 2 +k

+k shift up

Y = a(x-h) 2 -k

-k shift down

+a faces up

+a faces up

y = -4x2+2

y = -4x2-2

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

-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 = -16t2 + 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= -16t2+40 h

0 -16 (0)∙ 2+40 =40 40

-16 (1)∙ 2+40 =24 241

2 -16 (2)∙ 2+40 -24= -24

Notice:We stop after we get a negative height as we Cannot go beyond the ground.

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)

Heig

ht (h

)

VIDEOS: Quadratic Graphs and Their Properties

Interpreting Quadratics:http://www.khanacademy.org/math/algebra/quadratics/quadratic_odds_ends/v/algebra-ii--shifting-quadratic-graphs

http://www.khanacademy.org/math/algebra/quadratics/graphing_quadratics/v/graphing-a-quadratic-function

Graphing Quadratics:

VIDEOS: Quadratic Graphs and Their Properties

Graphing Quadratics:

http://www.khanacademy.org/math/algebra/quadratics/graphing_quadratics/v/quadratic-functions-1

CLASSWORK:

Page 537-539:

Problems: 1, 2, 3, 4, 7, 8, 10, 13, 19, 27, 28, 34 39.