4.2 Graphing Quadratic 4.2 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

• Changing between eqn. formsChanging between eqn. forms

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 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.

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