The Parabolic Bridge Project: Use of Parabolas in Bridge Design by Dr. K. VanFleet.
Parabola Formulas Parabolas (Type 2)Parabolas (Type 1) Vertex Form Vertex: (h, k) Axis: x = h...
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Transcript of Parabola Formulas Parabolas (Type 2)Parabolas (Type 1) Vertex Form Vertex: (h, k) Axis: x = h...
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