Section 2.3 beginning on page 68 Focus of a Parabola Section 2.3 beginning on page 68

The Big Ideas In this section we will learn about… The focus and the directrix of a parabola Writing equations for parabolas using the focus, the directrix and the distance formula. Recognizing equations of parabolas with a vertex at the origin, identifying the focus and directrix and axis of symmetry using the general formulas, and then graphing them. Writing equations for parabolas with a vertex at the origin using the general formulas. Standard Equations of parabolas with a vertex at (h,k). Writing Equations of parabolas with a vertex not on the origin.

Core Vocabulary Previous New Perpendicular Focus Distance Formula Congruent New Focus Directrix

The Focus and Directix A parabola can be defined as the set of all points (𝑥,𝑦) in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix.

Using the Distance Formula to Write an Equation Example 1: Use the Distance Formula to write an equation of the parabola with focus 𝐹(0,2) and directix 𝑦=−2. 𝑃𝐷=𝑃𝐹 (𝑥− 𝑥 1 ) 2 + (𝑦− 𝑦 1 ) 2 = (𝑥− 𝑥 2 ) 2 + (𝑦− 𝑦 2 ) 2 (𝑥−𝑥) 2 + (𝑦−(−2)) 2 = (𝑥−0) 2 + (𝑦−2) 2 (𝑦+2) 2 = 𝑥 2 + (𝑦−2) 2 Based on the definition of a parabola, the line segments drawn from F to P and from P to D are congruent. (𝑦+2) 2 = 𝑥 2 + (𝑦−2) 2 𝑦 2 +4𝑦+4= 𝑥 2 + 𝑦 2 −4𝑦+4 8𝑦= 𝑥 2 𝑦= 1 8 𝑥 2 𝑫= ( 𝒙 𝟐 − 𝒙 𝟏 ) 𝟐 + ( 𝒚 𝟐 − 𝒚 𝟏 ) 𝟐

Using the Distance Formula to Write an Equation 𝑃𝐷=𝑃𝐹 (𝑥− 𝑥 1 ) 2 + (𝑦− 𝑦 1 ) 2 = (𝑥− 𝑥 2 ) 2 + (𝑦− 𝑦 2 ) 2 (𝑥−𝑥) 2 + (𝑦−(3)) 2 = (𝑥−0) 2 + (𝑦−(−3)) 2 (𝑦−3) 2 = 𝑥 2 + (𝑦+3) 2 (𝑦−3) 2 = 𝑥 2 + (𝑦+3) 2 𝑦 2 −6𝑦+9= 𝑥 2 + 𝑦 2 +6𝑦+9 −12𝑦= 𝑥 2 𝑦=− 1 12 𝑥 2

General Equation of a Parabola With Vertex (0,0) These parabolas have a focus of (𝑝,0) and a directrix of 𝑥=−𝑝

Graphing an Equation of a Parabola Example 2: Identify the focus, directrix, and axis of symmetry of −4𝑥= 𝑦 2 . Graph the equation. Step 1: Write in standard form Step 2: Find focus, directrix, axis of sym Step 3: Find two points one side of the axis of sym, then reflect them to the other side of the axis of sym. −4𝑥= 𝑦 2 𝑥=− 1 4 𝑦 2 𝑥= 1 4𝑝 𝑦 2 Focus at (−1,0) Axis of sym is the x-axis Directrix 𝑥=1 𝑝=−1 𝑥=− 1 4 𝑦 2 𝑥=− 1 4 (1) 2 𝑥=−0.25 (−0.25,1) 𝑦=1 𝑦=2 𝑥=− 1 4 𝑦 2 𝑥=− 1 4 (2) 2 𝑥=−1 (−1,2)

Writing Equations of Parabolas Because the vertex is at the origin and the parabola opens down, the general equation is… The directrix is for this type of parabola is 𝑦=−𝑝, so … Substitute that value of p into the general equation.. 𝑦= 1 4𝑝 𝑥 2 𝑝=−3 𝑦= 1 4(−3) 𝑥 2 𝑦= 1 −12 𝑥 2 𝑦=− 1 12 𝑥 2

Quick Practice 𝑦= 1 6 𝑥 2 𝑥= 1 12 𝑦 2 𝑥=− 1 8 𝑦 2

Standard Equation of a Parabola With Vertex at (ℎ,𝑘)

Writing an Equation of a Translated Parabola Example 4: Write an equation of the parabola shown. This is a side facing parabola so the standard equation is… The Vertex is (ℎ,𝑘) The Focus is .. 𝑥= 1 4𝑝 (𝑦−𝑘) 2 +ℎ (6,2) (ℎ+𝑝,𝑘) (10,2) ℎ+𝑝=10 𝑥= 1 4𝑝 (𝑦−𝑘) 2 +ℎ ℎ=6 𝑘=2 𝑝=4 𝑥= 1 4(4) (𝑦−2) 2 +6 From vertex 𝑥= 1 16 (𝑦−2) 2 +6

Quick Practice 8) Write an equation of a parabola with vertex (−1, 4) and focus (−1, 2). If the vertex and the focus have the same x-coordinate, I know this is a parabola that opens up or down. (ℎ,𝑘) (ℎ,𝑘+𝑝) 𝑦= 1 4𝑝 (𝑥−ℎ) 2 +𝑘 Vertex = (−1,4) Focus = (−1,2) 𝑘+𝑝=2 ℎ=−1 𝑘=4 𝑝=−2 4+𝑝=2 𝑦= 1 4𝑝 (𝑥−ℎ) 2 +𝑘 𝑦= 1 4(−2) (𝑦−−1) 2 +4 𝑦=− 1 8 (𝑥+1) 2 +4

Solving a Real Life Problem

Quick Practice 9) A parabolic microwave antenna is 16 feet in diameter. Wire an equation that represents the cross section of the antenna with its vertex at (0,0) and its focus 10 feet to the right of the vertex. What is the depth of the antenna? With the vertex at the origin and the focus to the right of the vertex, we have a parabola that opens horizontally. General form of the equation: Because the focus is 10ft to the right of the vertex: Specific Equation: Since the antenna is 16 feet in diameter, we can find the depth by finding the x value when 𝑦=8 (the distance from the center of the antenna to the outer edge). 𝑥= 1 4𝑝 𝑦 2 𝑝=10 𝑥= 1 40 𝑦 2 𝑥= 1 40 𝑦 2 𝑥= 1 40 (8) 2 𝑥= 1 40 64 The antenna is 1.6 feet deep. 𝑥=1.6