5.1 Graphing Quadratic 5.1 Graphing Quadratic FunctionsFunctions

Do now:

Make up three examples of linear functions.

How do you know they are linear?

OBJ: to graph quadratic functions & use quadratic functions to solve real-life problems

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 makes a ≠0 makes a u-shaped graph called a u-shaped graph called a __________________..

Example quadratic equation:

parabola

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-Standard Form- y=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 atTo 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 an x-value & use the eqn to find the Choose an x-value & use the eqn to find the

corresponding y-value. Then use symmetry to corresponding y-value. Then use symmetry to find another pointfind another point

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

a

b

2

a

b

2

Ex 1Ex 1: Graph y = 2x: Graph y = 2x2 2 - 8x + 6- 8x + 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 22 -2-2

* Graph!* Graph!x=2

Graphing calculatorGraphing calculator• 22ndnd

• CALCCALC

• Option 3: Min or Option 4: MaxOption 3: Min or Option 4: Max• Left bound? Left bound? use your arrows until you are to the left of the use your arrows until you are to the left of the

vertexvertex

• EnterEnter• Right bound? Right bound? use your arrows until you are to the right of ituse your arrows until you are to the right of it

• EnterEnter

• EnterEnter

• Use 2Use 2ndnd TABLE to find two other points TABLE to find two other points

Practice:Practice:GraphGraph y = -xy = -x2 2 + x + 12+ x + 12

* Opens up or down?* Opens up or down?* What is the Vertex?* What is the Vertex?* Where is the Axis of symmetry?* Where is the Axis of symmetry?* Table of values with points & use * Table of values with points & use symmetrysymmetry

(-1,10)

(-2,6)

(2,10)

(3,6)

X = .5

(.5,12)

Vertex Form-Vertex Form-y = a(x - h)y = a(x - h)2 2 + 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 line The axis of symmetry is the vertical line

x=h.x=h.

• Don’t forget about 2 points on either side Don’t forget about 2 points on either side of the vertex!of the vertex!

Ex 2Ex 2:: Graph y= -.5(x + 3)Graph y= -.5(x + 3)2 2 + 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

-3 4-3 4

-5 2-5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

Graphing calculatorGraphing calculator

• 22ndnd

• TABLETABLE

• Find the “mirror” - Find the “mirror” - this is the vertexthis is the vertex

• Then choose 2 points - Then choose 2 points - one on either one on either sideside

Practice:Practice:

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

*Opens up or down?*Opens up or down?

*Vertex?*Vertex?

*Axis of symmetry?*Axis of symmetry?

*Table of values with points*Table of values with points

(-1, 11)

(0,5)

(1,3)

(2,5)

(3,11)

X = 1

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

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

2

qp 2

qp

Ex 3Ex 3: Graph y = -(x + 2)(x - 4): Graph y = -(x + 2)(x - 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=1vertical line x=1

x=1

(-2,0) (4,0)

(1,9)

Graphing calculatorGraphing calculator• 22ndnd

• CALCCALC• Option 2: “ zeros ”Option 2: “ zeros ”• Left bound? Left bound? use your arrows until you are to use your arrows until you are to

the left of the x-intthe left of the x-int

• EnterEnter• Right bound? Right bound? use your arrows until you are to use your arrows until you are to

the right of itthe right of it

• EnterEnter• EnterEnter• REPEAT for the other x-int (zero)REPEAT for the other x-int (zero)

Practice:Practice:

Graph y = 2(x - 3)(x + 1)Graph 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 use the distributive propertyThe key is to use the distributive property

Ex 4:Ex 4: y=-(x+4)(x-9) y=-(x+4)(x-9) Ex 5:Ex 5: y=3(x-1) 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+1)+8-x-x+1)+8

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

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

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