Warmup What is the radius and center of this circle? 𝑥 2 + 𝑦 2 + 12𝑥 − 10𝑦 −60=0

Write equations of parabolas. Graph parabolas. Supplement 3 Parabolas Write equations of parabolas. Graph parabolas.

Four Types of Conic Sections

Activity Animation

In previous chapters we worked with parabolas in general form, 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 and then completed the square to get them in vertex form 𝑦=𝑎 (𝑥−ℎ) 2 +𝑘 where the vertex (h, k) was easily found, and the sign of “a” determined whether it was opening up (positive) or down (negative). Now we will consider how to find the focus point and the directrix line from these equations.

For the simplest of these parabolas, p represents the focal distance (distance from vertex moving inside the curve to the focus). Going p units in the opposite direction will get you to the directrix. 𝑎= 1 4𝑝 𝑜𝑟 𝑝= 1 4𝑎 𝑎𝑥 2 =𝑦 If p (or a) is positive, equations with 𝑥 2 open up. If p (or a) is negative, they open down.

If we move the vertex to (h, k), the equation becomes 𝑦=𝑎(𝑥−ℎ) 2 +𝑘 Notice that a is in front of square-term . The relationship: 𝑎= 1 4𝑝 or 𝑝= 1 4𝑎 Latus rectum: the line segment perpendicular to line of symmetry thru focus, 4𝑝 in length.

Identify the vertex, focus and directrix of the parabola Identify the vertex, focus and directrix of the parabola. Then sketch the graph of the parabola. 4. 𝑦=− 1 4 𝑥 2

Parabolas can also open left or right. These come in the form a 𝑦 2 =𝑥 If p (or a) is positive, equations with 𝒚 𝟐 open right. If p (or a) is negative, they open left. p still represents the focal distance (distance from vertex moving inside the curve to the focus). Going p units in the opposite direction will get you to the directrix. With a vertex at (h, k), these become 𝑥= a (𝑦−𝑘) 2 +ℎ Notice that the vertex ordered pair comes from the outside for h, and inside with y for k

Identify the vertex, focus and directrix of the parabola Identify the vertex, focus and directrix of the parabola. Then sketch the graph of the parabola. 7. 𝑥=− 1 3 𝑦 2

Write each equation in standard form. 10. Vertex at origin, Focus ( 5 2 , 0)

𝑎= 1 4𝑝 or 𝑝= 1 4𝑎

Identify the vertex, focus and directrix of the equation Identify the vertex, focus and directrix of the equation. Then sketch the graph. 20. 𝑥=− 𝑦+6 2 −3