Parabola Formulas

Parabolas (Type 2)Parabolas (Type 1)

Vertex Form Vertex Form

khxay 2)( hkyax 2)(

Vertex: (h, k)

Axis: x = h

Vertex: (h, k)

Axis: y = k

Rate: a (+ up / –down) Rate: a (+ rt /–left)

Write each equation in vertex form. State the vertex and aos.

582 xxy

222 45)48( xxy

11)4( 2 xy

[1]

Vertex: (-4, -11)

Axis: x = -4

Write each equation in vertex form. State the vertex and aos.

1102 xxy

222 51)510( xxy

[2]

24)5( 2 xy

Vertex: (5, -24)

Axis: x = 5

Write each equation in vertex form. State the vertex and aos.

362 yyx

222 33)36( yyx

[3]

12)3( 2 yx

Vertex: (-12, 3)

Axis: y = 3

Write each equation in vertex form. State the vertex and aos.

132 yyx22

2

23

123

3

yyx

45

23 2

yx

[4]

Vertex:

Axis:

23

,45

23y

Example 1 Type 1 Parabolas

[A] 862 xxy [B] 342 xxy

1a

Vertex: (2, -1)

Axis: x = 2

1)2(

23)24(2

222

xy

xxy

Write in vertex form. Identify the vertex and axis of symmetry.

Example 2 Type 2 Parabolas Write in vertex form. Identify the vertex and axis of symmetry.

[A] 882 yyx [B] 462 yyx

1a

Vertex: (-5, 3)

Axis: y = 3

5)3(

34)36(2

222

yx

yyx

Write each equation in vertex form. State the vertex and aos.

1122 2 xxy

222 )3(21)36(2 xxy

[5]

Vertex: (-3, -19)

Axis: x = -3

19)3(2 2 xy

Write each equation in vertex form. State the vertex and aos.

742 2 xxy

222 )1(27)12(2 xxy

[6]

Vertex: (-1, 5)

Axis: x = -1

5)1(2 2 xy

Example 3 Type 1 Parabolas

[A] 50243 2 xxy [B] 322 xxy

1a

Vertex: (-1, 4)

Axis: x = -1

4)1(

13)12(2

222

xy

xxy

Write in vertex form. Identify the vertex and axis of symmetry.

Example 4 Type 2 Parabolas

[A] 7255 2 yyx [B] 1123 2 yyx

1a

Vertex: (13, -2)

Axis: y = -2

13)2(3

121)24(32

22

yx

yyx

Write in vertex form. Identify the vertex and axis of symmetry.

[1]

[2]

[3]

[4]

Write each equation in vertex form. State the vertex and aos.

582 xxy

1102 xxy

362 yyx

132 yyx

[5]

[6]

CW 8.2-8.3 (Day 1)

1122 2 xxy

742 2 xxy

Write each equation in vertex form. State the vertex and aos.

Example 3 Equation given Vertex & Point

[A] Type IVertex: (2, 4)Point: (-6, 8)

[B] Type II Vertex: (- 4, 6)Point: (2, 8)

khxay 2)( hkyax 2)(4)6( 2 yax

4)68(2 2 a

2

3

46

a

a

4)6(2

3 2 yx

Recall:

• Midpoint formula:

• Distance formula:

2,

22121 yyxx

221

221 )()( yyxxd

Conic Formulas

Circles Standard Form

222 )()( rkyhx

Center: (h, k)

Radius: r

Practice

1. Identify the center and radius.

2.Write an equation of the circle with a center (-1, 3) and radius of 6.

16)5()2( 22 yx

3.Write the equation of a circle if the endpoints are (-1, 7) and (5, -1)

4. Write an equation of a circle if the endpoints are (-3, -5) and (6, 2)

Practice

Example 5 Equations given the Diameter

[A] (5, 4) and (-2, -6)

Write the equation of the circle given the endpoints of a diameter.

[B] (2, 8) and (2, -2)

Center: (2, 3)

222 )3()2( ryx 222 )38()22( r

225 r

25)3()2( 22 yx

Writing in standard form

______29___)12(___)8( 22 yyxx

361629)3612()168( 22 yyxx

02912822 yxyx

Center: (-4,6)

Radius: 9

Complete the square for x’s and y’s

81)6()4( 22 yx

Example 6 Writing Circles in Standard Form

[A]

Write in standard form, find the radius and center. Graph.

[B]058422 yxyx 07622 xyx

16)3(

37)36(22

2222

yx

yxx

Center: (3, 0)

Radius: r = 4

Example 7 Writing Circles in Standard Form

[A]

Write in standard form, find the radius and center. Graph.

[B]058422 yxyx 07622 xyx

16)3(

37)36(22

2222

yx

yxx

Center: (3, 0)

Radius: r = 4

Example 8 Equations given the a Tangent

[A] Center: (-4, -3)tangent to x-axis

Write the equation of the circle given its tangency to an axis.

[B] Center: (3, 5)tangent to y-axis

9)5()3( 22 yx

Write each equation in standard form. State the center and radius.

098622 yxyx

222222 439)48()36( yyxx

[9]

Center: (-3, -4)

Radius: r = 4

16)4()3( 22 yx

Write each equation in vertex form. State the vertex.

06416822 yxyx

222222 8464)816()48( yyxx

[10]

Center: (4, -8)

Radius: r = 4

16)8()4( 22 yx