Objectives & HW Students will be able to identify vertex, focus, directrix, axis of symmetry, opening and equations of a parabola. HW: p. 403: 57-64 all

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 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 7.2 – Parabolas Parabola – 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. Axis of Symmetry Directrix - the fixed line of a parabola. Axis of Symmetry – The line that goes through the focus and is perpendicular to the directrix. Focus Latus Rectum Vertex – the point of intersection of the axis of symmetry and the parabola. Latus Rectum – the line segment through the focus and parallel to the directrix. Vertex Directrix

Equations and Graphs of Parabolas Section 7.2 – Parabolas Equations and Graphs of Parabolas Equation Vertex Focus Directrix Description 𝑦 2 =4𝑎𝑥 (0,0) (𝑎,0) 𝑥=−𝑎 𝑆𝑦𝑚:𝑥−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑟𝑖𝑔ℎ𝑡 𝑦 2 =−4𝑎𝑥 (0,0) (−𝑎,0) 𝑥=𝑎 𝑆𝑦𝑚:𝑥−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡

Equations and Graphs of Parabolas Section 7.2 – Parabolas Equations and Graphs of Parabolas Equation Vertex Focus Directrix Description 𝑥 2 =4𝑎𝑦 (0,0) (0,𝑎) 𝑦=−𝑎 𝑆𝑦𝑚:𝑦−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑢𝑝 𝑥 2 =−4𝑎𝑦 (0,0) (0,−𝑎) 𝑦=𝑎 𝑆𝑦𝑚:𝑦−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛

Equations and Graphs of Parabolas Section 7.2 – Parabolas Equations and Graphs of Parabolas Equation Vertex Focus Directrix Description (𝑦−𝑘) 2 =4𝑎(𝑥−ℎ) (ℎ,𝑘) (ℎ+𝑎,𝑘) 𝑥=ℎ−𝑎 𝑆𝑦𝑚: 𝑡𝑜 𝑥−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑟𝑖𝑔ℎ𝑡 (𝑦−𝑘) 2 =−4𝑎(𝑥−ℎ) (ℎ,𝑘) (ℎ−𝑎,𝑘) 𝑥=ℎ+𝑎 𝑆𝑦𝑚: 𝑡𝑜 𝑥−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡

Equations and Graphs of Parabolas Section 7.2 – Parabolas Equations and Graphs of Parabolas Equation Vertex Focus Directrix Description (𝑥−ℎ) 2 =4𝑎(𝑦−𝑘) (ℎ,𝑘) (ℎ,𝑘+𝑎) 𝑦=𝑘−𝑎 𝑆𝑦𝑚: 𝑡𝑜 𝑦−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑢𝑝 (𝑥−ℎ) 2 =−4𝑎(𝑦−𝑘) (ℎ,𝑘) (ℎ,𝑘−𝑎) 𝑦=𝑘+𝑎 𝑆𝑦𝑚: 𝑡𝑜 𝑦−𝑎𝑥𝑖𝑠 𝑂𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛

Section 7.2 – Parabolas Find the vertex, focus, directrix and the latus rectum for each equation 𝑥 2 =16𝑦 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎, 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝  𝑣𝑒𝑟𝑡𝑒𝑥:(0,0) 𝑓𝑖𝑛𝑑 𝑎  16=4𝑎 𝑎=4 𝑦=−4 𝑓𝑜𝑐𝑢𝑠 (0,0+𝑎) 𝑙𝑎𝑡𝑢𝑠 𝑟𝑒𝑐𝑡𝑢𝑚 (0,4) 𝑥 2 =16𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 𝑥 2 =16(4) 𝑦=0−𝑎 𝑥 2 =64 𝑦=−4 𝑥=±8 (−8,4) (8,4)

Find the equation given the focus (0, -2) and the directrix, x = 2 Section 7.2 – Parabolas Find the equation given the focus (0, -2) and the directrix, x = 2 𝑓𝑖𝑛𝑑 𝑎 𝑎= 2+0 2 𝑎=1   𝑣𝑒𝑟𝑡𝑒𝑥 (1,−2) 𝑥=2 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡 (𝑦−𝑘) 2 =−4𝑎(𝑥−ℎ) (𝑦−−2) 2 =−4(1)(𝑥−1) (𝑦+2) 2 =−4(𝑥−1)

Find the equation given the vertex (3, 1) and the focus (3, 5) Section 7.2 – Parabolas Find the equation given the vertex (3, 1) and the focus (3, 5)  𝑓𝑖𝑛𝑑 𝑎 𝑎=5−1  𝑎=4 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (𝑥−𝑘) 2 =4𝑎(𝑦−ℎ) 𝑦=−3 (𝑥−3) 2 =4(4)(𝑦−1) (𝑥−3) 2 =16(𝑦−1) 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 𝑦=1−4 𝑦=−3

Section 7.2 – Parabolas Find the vertex and the focus given: 𝑦 2 +10𝑦+𝑥+20=0 𝑦 2 +10𝑦+𝑥+20=0 𝑣𝑒𝑟𝑡𝑒𝑥 𝑦 2 +10𝑦=−𝑥−20 (5,−5) 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑖𝑛𝑑 𝑎 10 2 =5 5 2 =25 1=4𝑎 𝑎= 1 4 𝑦 2 +10𝑦+25=−𝑥−20+25 (𝑦+5) 2 =−𝑥+5 𝑓𝑜𝑐𝑢𝑠 (𝑦+5) 2 =−(𝑥−5) (5− 1 4 ,−5) 𝑜𝑝𝑒𝑛𝑠 𝑙𝑒𝑓𝑡 (4 3 4 ,−5)  