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6.6 Analyzing Graphs of Quadratic Functions Goal 1: Analyze quadratic functions of the form y=a(x-h) 2 +k Goal 2: Write a quadratic function in the form y=a(x-h) 2 +k

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Vertex form: y=a(x-h) 2 +k –(h,k): the vertex of the parabola –x=h: the axis of symmetry Remember: –adding inside the ( ) moves the graph to the left –subtracting inside the ( ) moves the graph to the right –adding outside the ( ) moves the graph up –subtracting outside the ( ) moves the graph down –multiplying by a whole number outside the ( ) makes the graph narrower –multiplying by a fraction outside the ( ) makes the graph narrower

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Ex. Analyze. Then draw the graph. y=(x+2) 2 +1 y=a(x-h) 2 +k y=(x-(-2)) 2 +1 h=-2, k=1 vertex: (-2, 1) Axis of symmetry: x=-2 Opens: Up This graph shifts left 2 places and up 1 place.

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Ex. Analyze. Then draw the graph. y=(x-3) 2 +2 y=a(x-h) 2 +k y=(x-3) 2 +2 h=3, k=2 vertex: (3, 2) Axis of symmetry: x=3 Opens: Up This graph shifts right 3 places and up 2 places.

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Ex. Write the function in vertex form. Then analyze the function. y=x 2 +8x-5 y=(x 2 +8x+c)-5-c y=(x 2 +8x+4 2 )-5-16 y=(x+4) 2 -21 y=(x-(-4)) 2 +(-21) Vertex: (-4, -21) Sym: x=-4 Opens: up This graph shifts left 4 places and down 21 places.

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Ex. Write the function in vertex form. Then analyze the function. y=x 2 +2x+4 y=(x 2 +2x+c)+4-c y=(x 2 +2x+1 2 )+4-1 y=(x+1) 2 +3 y=(x-(-1)) 2 +3 Vertex: (-1, 3) Sym: x=-1 Opens: up This graph shifts left 1 place and up 3 places.

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Ex. Write the function in vertex form. Then analyze the function. y=-3x 2 +6x-1 y =(-3x 2 +6x)-1 y =-3(x 2 -2x)-1 y =-3(x 2 -2x+c)-1-(-3)c Y = -3(x 2 -2x+1)-1-(-3)(1) y =-3(x-1) 2 -1-(-3)(1) y =-3(x-1) 2 -1+3 y =-3(x-1) 2 +2 Vertex: (1, 2) Sym: x=1 Opens: down This graph shifts right 1 place and up 2 places. This graph gets more narrow.

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Ex. Write the function in vertex form. Then analyze the function. y=-2x 2 -4x+2 y =(-2x 2 -4x)+2 y =-2(x 2 +2x)+2 y =-2(x 2 +2x+c)+2-(-2)c y =-2(x+1) 2 +2-(-2)(1) y =-2(x+1) 2 +2+2 y =-2(x+1) 2 +4 Vertex: (-1, 4) Sym: x=-1 Opens: down This graph shifts left 1 place and up 4 places. This graph gets wider.

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Ex. Write an equation for the parabola whose vertex is at (-1, 4) and passes through (2, 1). y=a(x-h) 2 +k (1)=a((2)-(-1)) 2 +(4) 1=a(2+1) 2 +4 -3=a(3) 2 -3=9a -1/3=a y=-1/3(x+1) 2 +4

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Ex. Write an equation for the parabola whose vertex is at (1, 2) and passes through (3, 4). y=a(x-h) 2 +k (4)=a((3)-(1)) 2 +(2) 4=a(3-1) 2 +2 2=a(2) 2 2=4a 1/2=a y=1/2(x-1) 2 +2

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