ANSWER KEY for Practice Quiz on Sections 9-1 through 9-5
(Quiz to be given Friday, 3/5/99)
1. Transform the following equation of a circle into standard form, and find the center and radius:
2x^2 + 20x + 2y^2 - 8y = 14
-- transformed eqn. (x+5)^2 + (y-2)^2 = 36
-- center at (-5,2)
-- r=6
2. Transform the following equation of an ellipse into standard form, and find the center, x-radius, y-radius, and focal radius. Sketch the graph, showing the coordinates of center and foci.
4x^2 + 16x + y^2 = 33
-- transformed eqn. ((x+2)/3.5)^2 + (y/7)^2 = 1
-- center at (-2,0)
-- rx (x-radius) = 3.5
-- ry (y-radius) = 7
-- c (focal radius) = sqrt(147)/2, or approx. 6.062.
-- Sketch should show an ellipse taller than it is wide, centered at (-2,0), with foci at (-2, 6.062) and (-2, -6.062). The ellipse must pass through the points (-2,7), (-2,-7), (-5.5,0), and (1.5,0).
3. Transform the following equation of a hyperbola into standard form, and find the center, x-radius, y-radius, and focal radius. Sketch the graph, showing the coordinates of center, vertices, and foci, as well as the asymptotes.
x^2 + 2x - 3y^2 + 18y = 2
-- transformed eqn. ((y-3)/sqrt(8))^2 - ((x+1)/sqrt(24))^2 = 1
-- center at (-1,3)
-- rx (x-radius) = sqrt(24), or approx. 4.9
-- ry (y-radius) = sqrt(8), or approx. 2.8
-- c (focal radius) = sqrt(32), or approx. 5.66
-- Sketch should show a hyperbola opening upward and downward, with center at (-1,3), vertices at (-1,5.8) and (-1,0.2), and foci at (-1,8.66) and (-1,-2.66). The asymptotes cross at the center, i.e., at (-1,3), and have slopes of plus or minus 1/sqrt(3).
4. The equation of a certain parabola can be written in the form 2x^2 + 4x + 2y + 8 = 15.
(a) Transform the parabola's equation into the "vertex form" that we learned last fall.
(b) Give the coordinates of the vertex.
(c) Does the parabola open upward, downward, left, or right? How can you tell?
(d) Sketch the graph, showing coordinates of all x- and y-intercepts.