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9.2: QUADRATIC FUNCTIONS:

Quadratic Function: A function that can be written in the form y = ax 2 +bx+c where a ≠ 0. 9.2: QUADRATIC FUNCTIONS:. 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 .

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9.2: QUADRATIC FUNCTIONS:

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  1. Quadratic Function: A function that can be written in the form y = ax2+bx+cwhere a ≠ 0. 9.2: QUADRATIC FUNCTIONS: Standard Form of a Quadratic: A function written in descending degree order, that is ax2+bx+c.

  2. 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.

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

  4. GOAL:

  5. To find the vertex of a quadratic equation where a ≠ 1, we must know the following: 1) = axis of symmetry . FINDING THE VERTEX OF ax2+bx+c: 2) = the x value of the vertex of the parabola. 3) y( ) = the y value of the vertex of the parabola, that is; min or max

  6. Ex:Provide the graph, vertex, domain and range of: y = 3x2-9x+2

  7. To graph we can create a table or we can find the vertex with other two points  Faster. y = 3x2-9x+2 SOLUTION: 1) = axis of symmetry . a = 3, b = -9    1.5 Thus x = 1.5 = axis of symmetry

  8. 2) = x value of the vertex  ( 1.5, ) 3) y( ) = the y value of the vertex of the parabola SOLUTION: (Continue) y = 3x2-9x+2 y = 3()2-9()+2  y = 3(1.5)2-9(1.5)+2  y = 3(2.25)-9(1.5)+2  y = 6.75-13.5+2  y= -4.75 Vertex = ( 1.5, -4.75)

  9. Vertex = ( 1.5, -4.75) Now: choose one value of x on the left of 1.5:  x=0 SOLUTION: (Continue) x = 0  3(0)2-9(0)+2  y = 2  (0, 2) choose a value of x on the right of 1.5:  x=3 x = 3  3(3)2-9(3)+2  y = 2  (3, 2) We not plot our three point: (0,2), vertex(1.5, -4.75) and (3,0)

  10. Vertex: ( 1.5, -4.75) (0, 2) SOLUTION: (3, 2) Domain: (-∞, ∞) Range: (-4.75, ∞)

  11. REAL-WORLD: During a basketball game halftime, the heat uses a sling shot to launch T-shirts at the crowd. The T-shirt is launched with an initial velocity of 72 ft/sec. The T-shirt is caught 35 ft above the court. How long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function that models the height of the T-shirt over time?

  12. To solve vertical motion problems we must know the following: An object projected into the air reaches the following maximum height: SOLUTION: h= -16 t2 + vt + c Where t = time, v = initial upward velocity c = initial height.

  13. During ….The T-shirt is launched with an initial velocity of 72 ft/sec. The T-shirt is caught 35 ft above the court. From the picture we can see thatinitial height is 5ft. Thus t = unknow,  v = initial upward velocity = 72 ft/sec c = initial height= 5 ft SOLUTION: h= -16 t2 + vt + c  h= -16 t2 + 72t + 5 Now: t =  t =  a = -16, b = 72  t =  t = 2.25

  14. 2) = 2.25 = x value of the vertex: ( 2.25, ) 3) y( ) = the y value of the vertex: SOLUTION: (Continue) y = -16t2+72t+5 y = -16()2+72()+5 y = -16(2.25)2+72(2.25)+5 y = -16(5.06)+72(2.25)+5  y = -81+162+5  y= 86 Vertex = ( 2.25, 86)

  15. Vertex = ( 2.25, 86) Now: choose one value of x on the left of 1.5:  x=0 SOLUTION: (Continue) x = 0  y = 5  (0, 5)  -16(0)2+72(0)+5 choose a value of x on the right of 1.5:  x=3 x = 4.5  -16(4.5)2+72(4.5)+5  (4.5, 5)  y = We not plot our three point: (0,5), vertex(2.25, 86) and (4.5, 5)

  16. Vertex: ( 2.25, -86) (0, 5) SOLUTION: (4.5, 5) Domain: (-∞, ∞) Range: (-∞, 86) Maximum : 86ft.

  17. 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/quadratic_odds_ends/v/algebra-ii--shifting-quadratic-graphs

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

  19. CLASSWORK:Page 544-545: Problems: 1, 2, 3, 4, 5, 8, 10, 13, 1619, 23, 25, 26.

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