math i

Math I

UNIT QUESTION: What is a quadratic function?Standard: MM2A3, MM2A4

Today’s Question:How do you graph quadratic functions in vertex form?Standard: MM2A3.b.

6.6 Graphing Quadratic 6.6 Graphing Quadratic FunctionsFunctions in Vertex or in Vertex or

Intercept FormIntercept Form

• DefinitionsDefinitions

• 3 Forms3 Forms

• Steps for graphing each formSteps for graphing each form

• ExamplesExamples

Daily Check

1.Factor: 3x2 + 10x + 8

2.Factor and Solve: 2x2 - 7x + 3 = 0

Parent Name: Quadratic

Parent Equation: 2y x x y

-2

-1

0

1

2

Domain: Range : 0y

General Equation(in Vertex Form)

2( )y a x h k

Domain:

Range : if 0y k a if 0y k a

Vertex: (0,0)

0x Axis of Symmetry:

Vertex: ,h k x hAxis of Symmetry:

4

1

0

4

1

Review: Quadratic Parent

AAT-A Date: 2/17/14 SWBAT write quadratic equations in vertex form. Do Now: Review Questions pg 336 #1-50, evensHW Requests: pg 303 #42-49; Pg 310 #15-37 odds

Worksheets for homework Skills Practice pg 340

Worksheet Quadratics Graphing TBDHW: Skills Practice Vertex form 6.6Announcements:

Life Is Just A MinuteLife is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!

By Dr. Benjamin Elijah Mays, Past President of Morehouse College

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=-Zil-EhDdjZjFM&tbnid=wAsYthktIKPgMM:&ved=0CAUQjRw&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBenjamin_Mays&ei=IW7XUq7kCu2MyAGXyIDICg&bvm=bv.59568121,d.aWc&psig=AFQjCNHlfFDR8M0n05e_wVH54vSgQ4E4tw&ust=1389936476306860

Writing a Quadratic Writing a Quadratic Function in Vertex FormFunction in Vertex Form

5. Solve for 5. Solve for yy the equation will be in the equation will be in vertex form.vertex form.

Steps:1. Write the function in standard form.

2. Set it up to complete the square.

3. Add the square to both sides of the = sign.

4. Write the trinomial as a binomial squared.

Writing a Quadratic Function in Vertex Writing a Quadratic Function in Vertex FormFormExample 1: Write the function in vertex Example 1: Write the function in vertex

form and identify its vertex. form and identify its vertex.

2 18 95f x x x

1. Write the function in standard form.2 18 95y x x


Writing a Quadratic Function in Vertex Writing a Quadratic Function in Vertex FormForm




29 14y x Vertex: 9, 14

Practice: Write a quadratic function in Practice: Write a quadratic function in vertex form and identify its vertex.vertex form and identify its vertex.P1:P1: 2 14 11f x x x

27 38y x

2 14 11y x x

7, 38vertex

Practice: Write a quadratic function in Practice: Write a quadratic function in vertex form and identify its vertex.vertex form and identify its vertex.P2:P2: 2 8 10g x x x

24 6y x 4, 6vertex

Writing a Quadratic Function in Vertex Writing a Quadratic Function in Vertex FormFormExample 2: Factor, write the function in Example 2: Factor, write the function in

vertex form, and identify its vertex.vertex form, and identify its vertex.

22 4 5f x x x 1. Write the function in standard form.

22 4 5y x x


2. Factor the first two terms.

22 2 5y x x





22 1 7y x Vertex: 1, 7


25 5 3y x 5, 3vertex

Practice: Write a quadratic function in Practice: Write a quadratic function in vertex form and identify its vertex.vertex form and identify its vertex.P4:P4:

Vertex:


1. Write the function in standard form.


Example 3: Write the function, using fractions, in vertex form, and identify its vertex.

Writing a Quadratic Function in Vertex Writing a Quadratic Function in Vertex FormForm3. Add the square to both sides of the = sign.Look! Be careful with the added term when a<1



Vertex:


29 75

22 2

y x

9 75,

2 2

Vertex:

Independent PracticeIndependent PracticeWrite each function in vertex Write each function in vertex

form and identify its vertex.form and identify its vertex.

21. 24 145f x x x

22. 3 18 7g x x x

23. 2 14 3h x x x

Quadratic FunctionQuadratic Function•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:

Vertex-Vertex-

• The lowest or highest pointThe lowest or highest point

of a parabola.of a parabola.

VertexVertex

Axis of symmetry-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.

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

Vertex FormVertex FormEach function we just looked at can be written Each function we just looked at can be written

in the form (x – h)in the form (x – h)22 + k, where (h , k) is the + k, where (h , k) is the vertex of the parabola, and x = h is its axis of vertex of the parabola, and x = h is its axis of symmetry.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, such as (-1, 2) and (0 , 5).side, 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.

Your Turn!Your Turn!

•Analyze and Graph:Analyze and Graph:

y = (x + 4)y = (x + 4)22 - 3. - 3.

(-4,-3)

Example 2: GraphExample 2: Graphy= -.5(x+3)y= -.5(x+3)22+4+4• a is negative (a = -.5), so parabola opens down.a is negative (a = -.5), so parabola opens down.• Vertex is (h,k) or (-3,4)Vertex is (h,k) or (-3,4)• Axis of symmetry is the vertical line x = -3Axis of symmetry is the vertical line x = -3• 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!

y=2(x-1)y=2(x-1)22+3+3

• Open up or down?Open up or down?

• Vertex?Vertex?

• Axis of symmetry?Axis of symmetry?

•Table of values with 4 points (other Table of values with 4 points (other than the vertex?than the vertex?

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

Intercept Form EquationIntercept Form Equationy=a(x-p)(x-q)y=a(x-p)(x-q)

• The x-intercepts are the points (p,0) and The x-intercepts are the points (p,0) and (q,0).(q,0).

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

• The x-coordinate of the vertex isThe x-coordinate of the vertex is

• To find the y-coordinate of the vertex, plug To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y.the x-coord. into the equation and solve for y.

• 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.

2

qp 2

qp

Example 3: Graph y=-(x+2)(x-Example 3: Graph y=-(x+2)(x-4)4)• Since a is negative, Since a is negative,

parabola opens parabola opens down.down.

• The x-intercepts are The x-intercepts are (-2,0) and (4,0)(-2,0) and (4,0)

• To find the x-coord. To find the x-coord. of the vertex, useof the vertex, use

• To find the y-coord., To find the y-coord., plug 1 in for x. plug 1 in for x.

• Vertex (1,9)Vertex (1,9)

2

qp

12

2

2

42

x

9)3)(3()41)(21( y

•The axis of The axis of symmetry is the symmetry is the vertical line x=1 vertical line x=1 (from the x-coord. (from the x-coord. of the vertex)of the vertex)

x=1

(-2,0) (4,0)

(1,9)

Now you try one!Now you try one!

y=2(x-3)(x+1)y=2(x-3)(x+1)

•Open up or down?Open up or down?

•X-intercepts?X-intercepts?

•Vertex?Vertex?

•Axis of symmetry?Axis of symmetry?

(-1,0) (3,0)

(1,-8)

x=1

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

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

AssignmentDay 1 -p. 65

#4,6,7,9,13,16

and Review for Quiz

Day 2 – p. 67 #4,5,7,9,11-14

We will not do intercept form.

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Documents

vertex formexample

vertex form3

vertex form5

vertex form4

vertex form1

skills practice vertex

quadratic functiona

standard form