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Warm-up!! Draw the circle (rough sketch) and give the center and radius. 1. ( x 2) 2 ( y 3) 2 36 Write an equation of a circle passing through the given point and has a center at the origin. 2. (8, 6) Write the equation of the circle in standard form. State the Center and Radius 3. x 2 y 2 14 x 2 y 49 0 CCGPS Geometry Day 60 (11-5-13) UNIT QUESTION: How are the equations of circles and parabolas derived? Standard: MCC9-12..A.REI.7, G.GPE.1,2 and 4 Today’s Question: How do we graph a parabola from a given equation in standard form? Standard: MCC9-12..G.GPE.2 Parabolas Parabolas Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix. The cross section of a headlight is an example of a parabola... Directrix The light source is the Focus Here are some other applications of the parabola... d2 d1Focus d3 d1 Vertex d3 d2 Directrix Notice that the vertex is located at the midpoint between the focus and the directrix... Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix... We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola.... Standard Equation of a Parabola: (Vertex at the origin) Equation 2 x = 4py Focus Directrix (0, p) y = –p (If the x term is squared, the parabola is up or down) Equation 2 y = 4px Focus Directrix (p, 0) x = –p (If the y term is squared, the parabola is left or right) Tell whether the parabola opens up down, left, or right. A. 2 y 5x B. 2 y 2 8x right C. 4x y2 left down Find the focus and equation of the directrix. Then sketch the graph. 1. y 16 x 2 4, 0 Focus : p,0 4 p 16 p4 Directrix : x p x 4 Opens right Find the focus and equation of the directrix. Then sketch the graph. 2. x 2 y 2 Focus : 0, p 4p 2 1 p 2 Directrix : y p Opens up 1 0, 2 1 y 2 Find the focus and equation of the directrix. Then sketch the graph. 3. x 12 y 2 Focus : 0, p 0, 3 4 p 12 p 3 Directrix : y p y 3 Opens down Find the focus and equation of the directrix. Then sketch the graph. 4. 3 y 12 x 0 2 Focus : 0, p 4 p 4 p 1 Directrix : y p x 1 1,0 Opens left Example 5: Determine the focus and directrix of the 2 parabola (y – 2) = -16 (x - 5) : Direction: Vertex: Focus: Directrix: Example 6: Determine the focus and directrix of the 2 parabola (x – 6) = 8(y + 3) : Direction: Vertex: Focus: Directrix: 7. Write the equation in standard form by completing the square. State the VERTEX. 2 x 2 x 8y 17 0 You TRY!! 8. Write the equation in standard form by completing the square. State the VERTEX. 2 y 6y 2x 9 0