This is the combination of slides from two people which

are

Melanie Tomlinson and Morrobea.

x

y

Focus (0,p)

Vertex

(h,k)

Geometric Definition of a Parabola: The collection of all the points P(x,y) in a a plane that are the same

distance from a fixed point, the focus, as they are from a fixed line called the directrix.

P

•Parabola is a Parabola is a Quadratic Quadratic functionfunction

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.

x

y

Focus (0,p)

Directrix

Vertex

(h,k)

p

p

2p

And the equation

is…

As you can plainly see the distance from the

focus to the vertex is a and is the same distance

from the vertex to the directrix! Neato!

y = -p

42 pyx

x

y

Focus (0,-p)

Directrix y a

Vertex

(h,k)

p

p

And the equation

is…

42 pyx

x

y

Focus (p,0)

Vertex

(h,k)

x aDirectrix p p

2p

And the equation

is…

42 pxy

x

y

Focus (-p,0)

Vertex

(h,k)

x aDirectrix p p

And the equation

is…

42 pxy

Your Turn!

• Analyze and Graph:

y = (x + 4)2 - 3.

(-4,-3)

Example 2: Graphy= -.5(x+3)2+4

• a is negative (a = -.5), so parabola opens down.• Vertex is (h,k) or (-3,4)• Axis of symmetry is the vertical line x = -3• Table of values x y

-1 2 -2 3.5

-3 4 -4 3.5 -5 2

Vertex (-3,4)

(-4,3.5)

(-5,2)

(-2,3.5)

(-1,2)

x=-3

Try this one!Try this 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

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaStandard Equation of a Parabola (Vertex at Origin)

pyx 42

p ,0focus

yx 122

3 ,0py

directrix3y

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaStandard Equation of a Parabola (Vertex at Origin)

pxy 42

0 ,pfocus

xy 122

0 ,3px

directrix

3x

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

yx 2 .1 2

focus:

2

1 ,0

directrix:2

1y

24 p2

1p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

xy 16 .2 2

focus: 0 ,4

directrix: 4x

164 p 4p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

yx4

1 .3 2

focus:

16

1 ,0

directrix:16

1y

4

14 p

16

1p

Graphing an Equation of a ParabolaGraphing an Equation of a ParabolaGraph the equation. Identify the focus and directrix of the parabola.

xy 4 .4 2

focus: 0 ,1

directrix: 1x

44 p 1p

1 ,0 .5

Writing an Equation of a ParabolaWriting an Equation of a ParabolaWrite the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

yx 42 pyx 42

1 yx 42

0 ,

2

1 .6

xy 42

pxy 42

2

1 xy 22

4

1 ,0 .8

Writing an Equation of a ParabolaWriting an Equation of a ParabolaWrite the standard form of the equation of the parabola with the given focus and vertex at (0, 0).

yx 42

pyx 42

4

1yx 2

0 ,2 .7

xy 42 pxy 42

2 xy 82

Modeling a Parabolic ReflectorModeling a Parabolic Reflector9. A searchlight reflector is designed so that a cross section through its axis is a parabola and the light source is at the focus. Find the focus if the reflector is 3 feet across at the opening and 1 foot deep.

1 ,5.1

ypx 42 25.1

4

25.2p

p425.2

400

22516

9

1

Notes Over 10.2Modeling a Parabolic ReflectorModeling a Parabolic Reflector10. One of the largest radio telescopes has a diameter of 250 feet and a focal length of 50 feet. If the cross section of the radio telescope is a parabola, find the depth.

yx 42 pyx 42

50yx 2002

2

250

y2001252

125y200625,15

ft 1.78y

General Form of any Parabola

2 2 0Ax By Cx Dy E

*Where either A or B is zero!

* You will use the “Completing the Square” method to go from the

General Form to Standard Form,

Graphing a Parabola: Use completing the square to convert a general form equation to standard

conic form

y2 - 10x + 6y - 11 = 0

9 9y2 + 6y + = 10x + 11 + _____(y + 3)2 = 10x + 20

(y + 3)2 = 10(x + 2)

General form

Standard form

aka: Graphing form

(y-k)2 = 4p(x-h)

ReferencesReferences

• Melanie Tomlinson, M. 2012. Melanie Tomlinson, M. 2012. http://www.slideshare.net/melanieitomlinson/parabola-14893256?qid=170dff68-76ee-4690-8e83-6d8e3a4d33e6&v=qf1&b=&from_search=2. . Accessed on 06 March 2014Accessed on 06 March 2014

• Morrobea. 2013. Morrobea. 2013. http://www.slideshare.net/morrobea/52-solve-quadratic-equations-by-graphingvertex-and-intercept-form?qid=170dff68-76ee-4690-8e83-6d8e3a4d33e6&v=qf1&b=&from_search=8. . Accessed on 06 March 2014Accessed on 06 March 2014