Section 7.1 – Conics

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Section 7.1 – Conics Conics – curves that are created by the intersection of a plane and a right circular cone.

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Section 7.1 – Conics. Conics – curves that are created by the intersection of a plane and a right circular cone. Section 7.1 – Conics. Conics – curves that are created by the intersection of a plane and a right circular cone. Section 7.1 – Conics. - PowerPoint PPT Presentation

Transcript of Section 7.1 – Conics

Page 1: Section 7.1 – Conics

Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.

Page 2: Section 7.1 – Conics

Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.

Page 3: Section 7.1 – Conics

Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.

Page 4: Section 7.1 – Conics

Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.

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Section 7.2 – ParabolasParabola – set of points in a plane that are equidistant from a fixed point (d(F, P)) and a fixed line (d (P, D)).Focus - the fixed point of a parabola.Directrix - the fixed line of a parabola.

Axis of Symmetry

Directrix

Axis of Symmetry – The line that goes through the focus and is perpendicular to the directrix.

Focus

Vertex – the point of intersection of the axis of symmetry and the parabola.

VertexLatus Rectum – the line segment through the focus and parallel to the directrix.

Latus Rectum

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Section 7.2 – ParabolasEquations and Graphs of Parabolas

𝑦 2=4𝑎𝑥Equation Vertex Focus Directrix Description

(0 ,0) (𝑎 ,0) 𝑥=−𝑎 𝑆𝑦𝑚 :𝑥−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠𝑟𝑖𝑔 h𝑡

𝑦 2=−4 𝑎𝑥 (0 ,0) (−𝑎 ,0) 𝑥=𝑎 𝑆𝑦𝑚 :𝑥−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡

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Equation Vertex Focus Directrix Description

Section 7.2 – ParabolasEquations and Graphs of Parabolas

𝑥2=4𝑎𝑦 (0,0) (0 ,𝑎) 𝑦=−𝑎 𝑆𝑦𝑚 : 𝑦−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠𝑢𝑝

𝑥2=−4𝑎𝑦 (0,0) (0 ,−𝑎) 𝑦=𝑎 𝑆𝑦𝑚 : 𝑦−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛

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Equation Vertex Focus Directrix Description

Section 7.2 – ParabolasEquations and Graphs of Parabolas

(𝑦−𝑘)2=4 𝑎(𝑥−h) (h ,𝑘) (h+𝑎 ,𝑘) 𝑥=h−𝑎𝑆𝑦𝑚 :𝑡𝑜 𝑥−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠𝑟𝑖𝑔 h𝑡

(𝑦−𝑘)2=−4𝑎(𝑥−h) (h ,𝑘) (h−𝑎 ,𝑘) 𝑥=h+𝑎𝑆𝑦𝑚 :𝑡𝑜 𝑥−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡

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Equation Vertex Focus Directrix Description

Section 7.2 – ParabolasEquations and Graphs of Parabolas

(𝑥−h)2=4𝑎(𝑦−𝑘) (h ,𝑘) (h ,𝑘+𝑎) 𝑦=𝑘−𝑎𝑆𝑦𝑚 :𝑡𝑜 𝑦−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠𝑢𝑝

(𝑥−h)2=−4 𝑎(𝑦−𝑘) (h ,𝑘) (h ,𝑘−𝑎) 𝑦=𝑘+𝑎𝑆𝑦𝑚 :𝑡𝑜 𝑦−𝑎𝑥𝑖𝑠𝑂𝑝𝑒𝑛𝑠𝑑𝑜𝑤𝑛

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Section 7.2 – ParabolasFind the vertex, focus, directrix and the latus rectum for each equation

𝑥2=16 𝑦

𝑣 𝑒𝑟𝑡𝑒𝑥 :(0 ,0)

16=4𝑎𝑎=4

𝑓 𝑖𝑛𝑑𝑎

𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 ,𝑜𝑝𝑒𝑛𝑠𝑢𝑝

𝑓 𝑜𝑐𝑢𝑠(0 ,0+𝑎)(0 ,4)

𝑦=0−𝑎

𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥

𝑦=−4

𝑦=−4

𝑙𝑎𝑡𝑢𝑠 𝑟𝑒𝑐𝑡𝑢𝑚𝑥2=16 𝑦𝑥2=16 (4 )𝑥2=64𝑥=±8

(−8 ,4 )(8 ,4)

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Section 7.2 – ParabolasFind the equation given the focus (0, -2) and the directrix, x = 2

𝑣 𝑒𝑟𝑡𝑒𝑥

𝑎=2+0

2𝑎=1

𝑓 𝑖𝑛𝑑𝑎

𝑜𝑝𝑒𝑛𝑠𝑙𝑒𝑓𝑡

𝑥=2 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛(𝑦−𝑘)2=−4𝑎(𝑥−h)

(1 ,−2)

(𝑦−−2)2=−4 (1)(𝑥−1)(𝑦+2)2=−4 (𝑥−1)

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Section 7.2 – ParabolasFind the equation given the vertex (3, 1) and the focus (3, 5)

𝑎=5−1𝑎=4

𝑓 𝑖𝑛𝑑𝑎

𝑜𝑝𝑒𝑛𝑠𝑢𝑝

𝑦=−3

𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛(𝑥−𝑘)2=4 𝑎(𝑦−h)

(𝑥−3)2=4 (4 )(𝑦−1)(𝑥−3)2=16 (𝑦−1)

𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥𝑦=1−4𝑦=−3

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Section 7.2 – ParabolasFind the vertex and the focus given:

1=4 𝑎𝑎=

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𝑓 𝑖𝑛𝑑𝑎

𝑜𝑝𝑒𝑛𝑠𝑙𝑒𝑓𝑡

𝑣 𝑒𝑟𝑡𝑒𝑥

𝑓 𝑜𝑐𝑢𝑠(5− 1

4 ,−5)

𝑦 2+10 𝑦+𝑥+20=0

𝑐 𝑜𝑚𝑝𝑙𝑒𝑡𝑒 h𝑡 𝑒𝑠𝑞𝑢𝑎𝑟𝑒𝑦 2+10 𝑦=−𝑥−20

102 =5 52=25

𝑦 2+10 𝑦+25=−𝑥−20+25(𝑦+5)2=−𝑥+5(𝑦+5)2=−(𝑥−5)

(5 ,−5)

(4 34 ,−5)