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

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.

Ex. Write the function in vertex form. Then analyze the function.

y=x2+8x-5y=(x2+8x+c)-5-cy=(x2+8x+42)-5-16y=(x+4)2-21y=(x-(-4))2+(-21)Vertex: (-4, -21)Sym: x=-4Opens: 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+4y=(x2+2x+c)+4-cy=(x2+2x+12)+4-1y=(x+1)2+3y=(x-(-1))2+3Vertex: (-1, 3)Sym: x=-1Opens: 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-1y =(-3x2+6x)-1y =-3(x2-2x)-1y =-3(x2-2x+c)-1-(-3)cY = -3(x2-2x+1)-1-(-3)(1)y =-3(x-1)2-1-(-3)(1)y =-3(x-1)2-1+3y =-3(x-1)2+2Vertex: (1, 2)Sym: x=1Opens: 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+2y =(-2x2-4x)+2y =-2(x2+2x)+2y =-2(x2+2x+c)+2-(-2)cy =-2(x+1)2+2-(-2)(1)y =-2(x+1)2+2+2y =-2(x+1)2+4Vertex: (-1, 4)Sym: x=-1Opens: down

This graph shifts left 1 place and up 4 places.

This graph gets wider.

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

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