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

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

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

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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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)

12. 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)

13. 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

14. 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)  