5.05a - Parabola at Origin | Lecture Note

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High School
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High School Physics
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2024
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5.05 THE PARABOLA Creating a Parabola • Created when the plane intersects one of the figures on an incline, no longer parallel, and passes through the base Definitions Parabola Focus Vertex Directrix Parabola • A locus of a point equidistant from a fixed point and a fixed line. Parabola Any point P on the locus will be equidistant from the focus (F) and a point (M) on the directrix such that 𝑃𝑀 ⊥ 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 𝑑 𝑃, 𝐹 = 𝑑(𝑃, 𝑀) Vertex The MIDPOINT of 𝑑𝐹 Where 𝑑 is the POI of the directrex and Focal Axis In other words: 𝑑𝑣 = 𝑣𝐹 𝑑𝑀 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 ⊥ 𝑑𝐹 (𝐹𝑜𝑐𝑎𝑙 𝐴𝑥𝑖𝑠) 5.05A CENTRED AT THE ORIGIN Where: 1) The vertex is halfway between the focus and the directrix 2) The focal axis is perpendicular to the directrix Horizontal Right 2 𝑦 = 4𝑐𝑥 𝑐>0 Where: Vertex 0, 0 Focus 𝑐, 0 Directrix: 𝑥 = −𝑐 𝑑 −𝑐, 0 Horizontal Left 2 𝑦 = −4𝑐𝑥 𝑐>0 Where: Vertex 0, 0 Focus −𝑐, 0 Directrix: 𝑥 = 𝑐 𝑑 𝑐, 0 Vertical Up 2 𝑥 = 4𝑐𝑦 𝑐>0 Vertex: (0,0) Focus: 𝐹(0, 𝑐) Directrix: 𝑦 = −𝑐 𝑑(0, −𝑐) Vertical Down 2 𝑥 = −4𝑐𝑦 𝑐>0 Vertex: (0,0) Focus: 𝐹(0, −𝑐) Directrix: 𝑦 = 𝑐 𝑑(0, 𝑐) Inequalities Inside 𝑥 2 ≤ 4𝑐𝑦 Outside 𝑥 2 > 4𝑐𝑦 𝑥 2 ≤ −4𝑐𝑦 𝑥 2 > −4𝑐𝑦 𝑦 2 < 4𝑐𝑥 𝑦 2 ≥ 4𝑐𝑥 𝑦 2 < −4𝑐𝑥 𝑦 2 ≥ −4𝑐𝑥 Proving the Rule • Start with concept of distance • 𝑑 𝑃, 𝐹 = 𝑑 𝑃, 𝑀 ∆𝑥 2 + ∆𝑦 2 = (𝑥 − 𝑐)2 +(𝑦 − 0)2 = ∆𝑥 2 + ∆𝑦 2 (𝑥 − −𝑐)2 +(𝑦 − 𝑦)2 Expand 𝑥 2 − 2𝑐𝑥 + 𝑐 2 + 𝑦 2 = 𝑥 2 + 2𝑐𝑥 + 𝑐 2 Since both sides are roots, square to eliminate 𝑥 2 − 2𝑐𝑥 + 𝑐 2 + 𝑦 2 = 𝑥 2 + 2𝑐𝑥 + 𝑐 2 Collect terms Graphing the Parabola 1) Determine the direction based on the equation 2) Label the Vertex 3) Find the value of c: 4) Plot F and d 5) Draw the Directrix 6) Draw the Parabola 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 Graphing Parabolas 2 Graph 𝑥 = 8𝑦 1) Determine the direction based on the equation UP 2) Label Vertex (0,0) 3) Find the value of c: 𝑥 2 = 4𝑐𝑦 4𝑐 = 8 c=2 4) Plot F and d F 0,2 and d(0, −2) 5) Draw the Directrix 𝑦 = −2 6) Draw the Parabola 2 Graph 𝑥 = 8𝑦 → c = 2 4) Plot F (0,2) and D (0, -2) 5) Draw the Directrix 6) Parabola consider the fact that the distance from the parabola to the Focus Point and to the Directrix must be the same. F(0, 2) V(0, 0) d(0, -2) Directrix y = −2 Finding the Rule 1) Consider the graph given, choose your rule based on the image. 2) Identify the Focal Point OR Point d 3) Find the distance from found point to the vertex. This is the value of c. 1) Consider the graph given, choose your rule based on the image. 𝑦 2 = 4𝑐𝑥 2) Identify the Focal Point OR Point d Given at (2, 0) 3) Find the distance from found point to the vertex. This is the value of c. 𝑦 2 = 4(2)𝑥 𝑦 2 = 8𝑥 1) Consider the graph given, choose your rule based on the image. 𝑥 2 = −4𝑐𝑦 2) Identify the Focal Point OR Point d Given at (0, -5) 3) Find the distance from found point to the vertex. This is the value of c. 𝑥 2 = −4 5 𝑦 𝑥 2 = −20𝑦 Homework • 5.05a – Parabola Centered at the Origin • Visions 2 • P193: 1ace, 2ace, 7ace, 9ae, 11bc