1033 CHAPTER 10 Test Prep Concepts Examples Equation Forms for Translated Parabolas A parabola with vertex 1h, k2 has an equation of the form 1 x −h22 =4p1 y −k2 Vertical axis of symmetry or 1 y −k22 =4p1x −h2. Horizontal axis of symmetry The focus is distance 0 p0 from the vertex. x y F(–1, 3) x = 3 V(1, 3) (y – 3)2 = –8(x – 1) y = 3 x y 3 –3 –2 2 0 + = 1 9 x2 4 y2 F9(–Ë5 , 0) F(Ë5 , 0) F9(0, – Ë5) F(0, Ë5) x y 2 –2 –3 3 0 + = 1 4 x2 9 y2 x y (3, 0) (3, 4) (3, 2) V9(–1, 2) V(7, 2) 0 + = 1 16 (x – 3)2 4 (y – 2)2 F9(3–2Ë3 , 2) F(3+2Ë3 , 2) x y = x y = –x y 2 2 –2 –2 0 – = 1 4 x2 4 y2 F9(–2Ë2, 0) F(2Ë2, 0) x y 2 2 –2 –2 0 – = 1 4 y2 4 x2 y = x y = –x F(0, 2Ë2) F9(0, –2Ë2) x y 3 2 (3, 2) V9(–1, 2) V(7, 2) 0 – = 1 16 (x – 3)2 4 (y – 2)2 asymptotes: y = ± (x – 3) + 2 1 2 10.2 Ellipses Standard Forms of Equations for Ellipses The ellipse with center at the origin and equation x2 a2 + y2 b2 =1 1where a 7b2 has vertices 1{a, 02, endpoints of the minor axis 10, {b2, and foci 1{c, 02, where c2 = a2 - b2. The ellipse with center at the origin and equation x2 b2 + y2 a2 =1 1where a 7b2 has vertices 10, {a2, endpoints of the minor axis 1{b, 02, and foci 10, {c2, where c2 = a2 - b2. Translated Ellipses The preceding equations can be extended to ellipses having center 1h, k2 by replacing x and y with x - h and y - k, respectively. 10.3 Hyperbolas Standard Forms of Equations for Hyperbolas The hyperbola with center at the origin and equation x2 a2 − y2 b2 =1 has vertices 1{a, 02, asymptotes y = { b a x, and foci 1 {c, 02, where c2 = a2 + b2. The hyperbola with center at the origin and equation y2 a2 − x2 b2 =1 has vertices 10, {a2, asymptotes y = { a b x, and foci 10, {c2, where c2 = a2 + b2. Translated Hyperbolas The preceding equations can be extended to hyperbolas having center 1h, k2 by replacing x and y with x - h and y - k, respectively.
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