Do Now

1. Factor: f(x) = 3x2 + 10x + 8

2. Factor f(x) = 2x2 - 7x + 3

TodayToday’’s Question:s Question:

Today’s Question:How do you graph quadratic functions in vertex form?

What important characteristics do you see in the vertex form?

Standard FormStandard Form•A function of the form A function of the form

y=axy=ax22+bx+c where a+bx+c where a≠0 making a ≠0 making a u-shaped graph called a u-shaped graph called a parabolaparabola..

Example quadratic equation:

Let’s ReviewLet’s Review

What is the Vertex?What is the Vertex?

•The lowest or highest pointThe lowest or highest point

of a parabola. of a parabola. VertexVertex

What is the Axis of Symmetry?What is the Axis of Symmetry?

•The vertical line through the vertex of the The vertical line through the vertex of the parabola.parabola.

Axis ofSymmetry

Vertex Form EquationVertex Form Equationy=a(x-h)y=a(x-h)22+k+k

• If a is positive, parabola opens upIf a is positive, parabola opens up

If a is negative, parabola opens down.If a is negative, parabola opens down.

• The vertex is the point (h,k).The vertex is the point (h,k).

• The axis of symmetry is the vertical The axis of symmetry is the vertical line x=h.line x=h.

• DonDon’’t forget about 2 points on either t forget about 2 points on either side of the vertex! (5 points total!)side of the vertex! (5 points total!)

Vertex FormVertex FormEvery function can be written in the form (x – Every function can be written in the form (x –

h)h)22 + k, where (h , k) is the vertex of the + k, where (h , k) is the vertex of the parabola, and x = h is its axis of symmetry.parabola, and x = h is its axis of symmetry.

(x – h)(x – h)22 + k – vertex form + k – vertex formEquationEquation VertexVertex Axis of Axis of

SymmetrySymmetry

y = xy = x22 or or y = (x – y = (x – 00))22 + + 00

((00 , , 00)) x = x = 00

y = xy = x22 + 2 or + 2 ory = (x – y = (x – 00))22 + + 22

((0 0 , , 22)) x = x = 00

y = (x – y = (x – 33))22 or or y = (x – y = (x – 33))22 + + 00

((33 , , 00)) x = x = 33

Example 1: Graph Example 1: Graph y = (x + 2)y = (x + 2)22 + 1 + 1•Analyze y = (x + 2)Analyze y = (x + 2)22 + 1. + 1.• Step 1 Step 1 Plot the vertex (-2 , 1)Plot the vertex (-2 , 1)

• Step 2 Step 2 Draw the axis of symmetry, x = -Draw the axis of symmetry, x = -2.2.

• Step 3Step 3 Find and plot two points on one Find and plot two points on one side side , such as (-1, 2) and (0 , 5)., such as (-1, 2) and (0 , 5).

• Step 4Step 4 Use symmetry to complete the Use symmetry to complete the graph, or find two points on thegraph, or find two points on the

• left side of the vertex.left side of the vertex.

With a partner: Find the key With a partner: Find the key characteristics: f(x) = characteristics: f(x) = -.5(x+3)-.5(x+3)22+4+4• Does parabola open up of down?Does parabola open up of down?• Vertex is (h,k) Vertex is (h,k) • Axis of symmetry x = Axis of symmetry x = • Table of values Table of values x y x y

-1 2-1 2 -2 3.5 -2 3.5

-3 4-3 4 -4 3.5-4 3.5 -5 2-5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

Now you try one!Now you try one!

Changing from vertex or Changing from vertex or intercepts form to standard intercepts form to standard

formform• The key is to FOIL! (first, outside, inside, The key is to FOIL! (first, outside, inside,

last)last)

• Ex: y=-(x+4)(x-9)Ex: y=-(x+4)(x-9) Ex: y=3(x-1)Ex: y=3(x-1)22+8+8

=-(x=-(x22-9x+4x-36)-9x+4x-36) =3(x-1)(x-1)+8 =3(x-1)(x-1)+8

=-(x=-(x22-5x-36)-5x-36) =3(x =3(x22-x--x-x+1)+8x+1)+8

y=-xy=-x22+5x+36+5x+36 =3(x =3(x22--2x+1)+82x+1)+8

=3x=3x22-6x+3+8-6x+3+8

y=3xy=3x22-6x+11-6x+11

Converting from standard to vertex Converting from standard to vertex fomfom

http://www.virtualnerd.com/algebra-2/quadratics/solve-by-completing-square-roots/complete-square/completing-square-convert-standard-to-vertex

Challenge Problem Challenge Problem

• Write the equation of the graph in vertex Write the equation of the graph in vertex form.form.

23( 2) 4y x

(-1,0) (3,0)

(1,-8)

x=1