1. 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

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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

10. 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