# 6.6 Analyzing Graphs of Quadratic Functions

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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

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

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.

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

Ex. Write the function in vertex form. Then analyze the function.
y=x2+8x-5 y=(x2+8x+c)-5-c y=(x2+8x+42)-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.

Ex. Write the function in vertex form. Then analyze the function.
y=x2+2x+4 y=(x2+2x+c)+4-c y=(x2+2x+12)+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.

Ex. Write the function in vertex form. Then analyze the function.
y=-3x2+6x-1 y =(-3x2+6x)-1 y =-3(x2-2x)-1 y =-3(x2-2x+c)-1-(-3)c Y = -3(x2-2x+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.

Ex. Write the function in vertex form. Then analyze the function.
y=-2x2-4x+2 y =(-2x2-4x)+2 y =-2(x2+2x)+2 y =-2(x2+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.

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
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

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
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