5.1 Graphing Quadratic 5.1 Graphing Quadratic FunctionsFunctions(p. 249)(p. 249)

• DefinitionsDefinitions

• 3 forms for a quad. function3 forms for a quad. function

• 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

Standard Form EquationStandard Form Equationy=axy=ax22 + bx + c + bx + c

• If a is If a is positivepositive, u opens , u opens upupIf a is If a is negativenegative, u opens , u opens downdown

• The x-coordinate of the vertex is atThe x-coordinate of the vertex is at• To find the y-coordinate of the vertex, plug the To find the y-coordinate of the vertex, plug the

x-coordinate into the given eqn.x-coordinate into the given eqn.• The axis of symmetry is the vertical line x=The axis of symmetry is the vertical line x=• Choose 2 x-values on either side of the vertex x-Choose 2 x-values on either side of the vertex x-

coordinate. Use the eqn to find the coordinate. Use the eqn to find the corresponding y-values. corresponding y-values.

• Graph and label the 5 points and axis of Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the symmetry on a coordinate plane. Connect the points with a smooth curve.points with a smooth curve.

a

b

2

a

b

2

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!)

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 1Example 1: Graph y=2x: Graph y=2x22--8x+68x+6• a=2 Since a is positive a=2 Since a is positive

the parabola will open the parabola will open up.up.

• Vertex: use Vertex: use b=-8 and a=2b=-8 and a=2

Vertex is: (2,-2)Vertex is: (2,-2)

a

bx

2

24

8

)2(2

)8(

x

26168

6)2(8)2(2 2

y

y

• Axis of symmetry is the Axis of symmetry is the vertical line x=2vertical line x=2

•Table of values for other Table of values for other points: points: x y x y

00 66 11 00 22 -2-2 33 00 44 66

* Graph!* Graph!x=2

Now you try one!Now you try one!

y=-xy=-x22+x+12+x+12

* Open up or down?* Open up or down?* Vertex?* Vertex?

* Axis of symmetry?* Axis of symmetry?* Table of values with 5 * Table of values with 5

points?points?

(-1,10)

(-2,6)

(2,10)

(3,6)

X = .5

(.5,12)

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 5 points?Table of values with 5 points?

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

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

Assignment