Download - Section 7.1 – Conics
Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.
Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.
Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.
Section 7.1 – ConicsConics – curves that are created by the intersection of a plane and a right circular cone.
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
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
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
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
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
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)
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)
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
Section 7.2 – ParabolasFind the vertex and the focus given:
1=4 𝑎𝑎=
14
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)