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3.2 Graphing Quadratic Functions in Vertex or Intercept Form Definitions Definitions 3 Forms 3 Forms Graphing in vertex form Graphing in vertex form Examples Examples Changing between eqn. forms Changing between eqn. forms

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Quadratic Function A function of the form y=ax 2 +bx+c where a0 making a u-shaped graph called a parabola. A function of the form y=ax 2 +bx+c where a0 making a u-shaped graph called a parabola. Example quadratic equation:

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Vertex- The lowest or highest point of a parabola. Vertex Axis of symmetry- The vertical line through the vertex of the parabola. Axis of Symmetry

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Vertex Form Equation y=a(x-h)2+k If a is positive, parabola opens up If a is negative, parabola opens down. The vertex is the point (h,k). The axis of symmetry is the vertical line x=h.

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Vertex Form Each function we just looked at can be written in the form (x – h) 2 + k, where (h, k) is the vertex of the parabola, and x = h is its axis of symmetry. Each function we just looked at can be written in the form (x – h) 2 + k, where (h, k) is the vertex of the parabola, and x = h is its axis of symmetry. (x – h) 2 + k – vertex form (x – h) 2 + k – vertex form EquationVertex Axis of Symmetry y = x 2 or y = (x – 0) (0, 0) x = 0 y = x or y = (x – 0) (0, 2) x = 0 y = (x – 3) 2 or y = (x – 3) (3, 0) x = 3

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Example 1: Graph Analyze y = (x + 2) Analyze y = (x + 2) Step 1 Plot the vertex (-2, 1) Step 1 Plot the vertex (-2, 1) Step 2 Draw the axis of symmetry, x = -2. Step 2 Draw the axis of symmetry, x = -2. Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 4 Use symmetry to complete the graph, or find two points on Step 4 Use symmetry to complete the graph, or find two points on the left side of the vertex. the left side of the vertex.

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Your Turn! Analyze and Graph: Analyze and Graph: y = (x + 4) y = (x + 4) (-4,-3)

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Example 2: Graph y=-.5(x+3) 2 +4 a is negative (a = -.5), so parabola opens down. a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Axis of symmetry is the vertical line x = -3 Table of values x y Table of values x y Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

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Now you try one! y=2(x-1) 2 +3 Open up or down? Open up or down? Vertex? Vertex? Axis of symmetry? Axis of symmetry? Table of values with 5 points? Table of values with 5 points?

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(-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

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Changing from vertex or intercepts form to standard form The key is to follow ORDER OF OPERATIONS The key is to follow ORDER OF OPERATIONS Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 y=-x 2 +5x+36 =3(x 2 -2x+1)+8 =3x 2 -6x+3+8 =3x 2 -6x+3+8 y=3x 2 -6x+11 y=3x 2 -6x+11

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Changing from vertex or intercepts form to standard form Practice: Practice: 1: y = 3(x-4)(x+2) 1: y = 3(x-4)(x+2) 2: y = -2(x-3) : y = -2(x-3) 2 - 5

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Challenge Problem Write the equation of the graph in vertex form. Write the equation of the graph in vertex form.

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Practice Workbook Page 68 #16-21

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Assignment Book Page 66 #25-33 and Page 68 #27, 28

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