Math 9.1 Pre-Calc Quiz Review

9.1-9.3 Quiz REVIEW HPC Name:_______________________________________ Hour:______ Find the standard form of the equation of the circle with the given characteristics. 1. Center: (0, 0); point on circle: (8, −15) 2. Endpoints of diameter: (−2, 3) and (6, −5) Write the equation of the circle in standard form. Sketch the circle and then identify its center, radius, x-intercepts, and y-intercepts. 3. 𝑥 2 + 𝑦 2 + 4𝑥 + 6𝑦 − 3 = 0 Center: Radius: x-intercepts: y-intercepts: 4. 4𝑥 2 + 4𝑦 2 + 32𝑥 − 24𝑦 + 51 = 0 Center: Radius: x-intercepts: y-intercepts: Write the equation of the parabola in standard form. Sketch the parabola and then identify its vertex, focus, and directrix. 5. 𝑦 2 − 4𝑦 − 4𝑥 = 0 Vertex: Focus: Directrix: 6. 𝑥 2 + 4𝑥 + 6𝑦 − 2 = 0 Vertex: Focus: Directrix: Find the standard form of the equation of the parabola with the given characteristics. 7. Vertex: (4, 2); Focus: (4, 0) 8. Focus: (8, −2); Directrix: 𝑥 = 4 9. A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1.25 feet deep at its center, determine the location of its focus. 10. A parabolic archway is 12 meters high at its vertex. At a height of 10 meters, the width of the archway is 8 meters. How wide is the archway at ground level? Write the equation of the ellipse in standard form. Sketch the ellipse and then identify its center, vertices, co-vertices, foci and eccentricity. 11. 4𝑥 2 + 25𝑦 2 + 16𝑥 − 150𝑦 + 141 = 0 Center: Vertices: Co-Vertices: Foci: Eccentricity: Co-Vertices: Foci: Eccentricity: 12. 16𝑥 2 + 9𝑦 2 − 32𝑥 + 72𝑦 + 16 = 0 Center: Vertices: 13. Find the standard form of the equation of the ellipse with the following characteristics: Center at (4, −2), Foci at (4, 1) and (4, −5), and minor axis of length 10. 14. A sunburst window above a doorway is constructed in the shape of a top half of a horizontally oriented ellipse. The window is 20 in. tall at its highest point, and 80 in. wide at the bottom. Find the height of the window, 25 in. from the center of the base. Write the equation of the hyperbola in standard form. Sketch the hyperbola and then identify its center, vertices, foci and equations for the asymptotes [leave these in point-slope form]. 15. 𝑦 2 − 9𝑥 2 = 9 Center: Vertices: Foci: Equations for Asymptotes: 16. 4𝑥 2 − 9𝑦 2 − 16𝑥 + 54𝑦 − 29 = 0 Center: Vertices: Foci: Equations for Asymptotes: 17. 9𝑥 2 − 𝑦 2 − 72𝑥 + 8𝑦 + 119 = 0 Center: Vertices: Foci: Equations for Asymptotes: Find the standard form of the equation of the hyperbola with the given characteristics. 18. Vertices: (0, 4), (0, 0), passes through (√5, −1) 19. Vertices: (3, 0), (3, −6), Asymptotes: 20. Foci at (2, 1) and (2, 7); and vertices (2, 6) and (2, 2) Identify the graph of each equation as a parabola, an ellipse, a circle or a hyperbola. 21. 4𝑥 2 + 9𝑦 2 − 16𝑥 − 36𝑦 + 16 = 0 22. 2𝑥 2 + 3𝑦 − 8𝑥 + 2 = 0 23. 5𝑥 − 4𝑦 2 + 24𝑦 − 11 = 0 24. 9𝑥 2 − 25𝑦 2 − 18𝑥 + 50𝑦 = 0 25. 4𝑥 2 + 4𝑦 2 − 4𝑥 + 8𝑦 − 11 = 0 26. 3𝑥 2 + 2𝑦 2 − 12𝑥 − 12𝑦 + 29 = 0 𝑦 =𝑥−6 𝑦 = −𝑥 Answers: 1. 𝑥 2 + 𝑦 2 = 289 3. (𝑥 + 2)2 + (𝑦 + 3)2 = 16; Cent.: (− 2, −3); Rad.: 4; 𝒙-int.: −2 ± √7; 𝒚-int.: −3 ± 2√3 4. (𝑥 + 4)2 + (𝑦 − 3)2 = 5. (𝑦 − 2)2 = 4(𝑥 + 1); Vertex.: (− 1, 2); Focus.: (0, 2); Directrix: 𝑥 = − 2 6. (𝑥 + 2)2 = −6(𝑦 − 1); Vertex.: (− 2, 1); Focus.: (−2, −0.5); Directrix: 𝑦 = 2.5 7. (𝑥 − 4)2 = −8(𝑦 − 2) 11. (𝑥 + 2)2 25 + (𝑦 − 3)2 4 (𝑥−1)2 9 + (𝑦+4)2 16 15. 16. 17. 18. 22. (𝑥 − 4)2 25 𝑦2 9 − 𝑥2 1 (𝑦 − 3)2 4 (𝑥−4)2 1 − 8 ± √13 ; 2 (𝑦 + 2)2 = 8(𝑥 − 6) 8. 𝒚-int.: None 9. 10. 3.2 𝑓𝑡 19.6 𝑚 √21 5 ≈ 0.92 (𝑦 + 2)2 34 √7 4 ≈ 0.66 14. =1 15.6 in. = 1; C: (0, 0); V: (0, ±3); F: (0, ±√10); Asy: 𝑦 = ±3𝑥 − − (𝑦 − 2)2 4 + 7 2 Cent.: (− 4, 3); Rad.: ; 𝒙-int.: = 1; C: (1, −4); V: (1, 0) and (1, −8); CV: (− 2, −4) and (4, −4); F: (1, −4 ± √7); E: 13. 49 ; 4 = 1; C: (− 2, 3); V: (3, 3) and (− 7, 3); CV: (− 2, 5) and (− 2, 1); F: (−2 ± √21, 3); E: 12. (𝑥 − 2)2 + (𝑦 + 1)2 = 32 2. (𝑥 − 2)2 9 (𝑦−4)2 − 9 𝑥2 4 Parabola 2 = 1; C: (2, 3); V: (2, 1) and (2, 5); F: (2, 3 ± √13); Asy: 𝑦 − 3 = ± 3 (𝑥 − 2) = 1; C: (4, 4); V: (3, 4) and (5, 4); F: (4 ± √10, 4); Asy: 𝑦 − 4 = ±3 (𝑥 − 4) =1 23. 19. (𝑦+3)2 9 Parabola − (𝑥 − 3)2 24. 9 =1 20. Hyperbola (𝑦 − 4)2 4 − (𝑥 − 2)2 25. 5 =1 Circle 21. Ellipse 26. Ellipse