Math 8 EXAM #5 Name: Complete all problems in your blue book. Copy the problem into the bluebook then show all of the required work for that problem. Work problems out down the page, not across. Make only one column of work per page. Skip a line between each problem. Any work or answers completed on this test form, other than the problems that require you to graph, will not be graded. For problems that REQUIRE YOU TO GRAPH use the graphs provided on this test form and complete the work for those problems in the space next to the graph. If you feel that you need more room, then indicate next to the graph that the work is in your bluebook. Complete, organized, and correct work, using the methods taught in class and including correct use of symbols, is required to receive full credit. Partial or no credit will be given when work is missing. Each problem is worth 5 points. SHOW ALL WORK. DO NOT ROUND ANSWERS UNLESS INSTRUCTED TO DO SO. Express answers in lowest terms. Simplify all radicals. Write an equation for the parabola. ) With vertex at (, ) and focus at (-5, ) p = -5 - = -8 a = (-8) = - x- = - y - Find an equation for the ellipse. ) Foci at (-, ) and (-, -); length of major axis is a = c = - [-]) a = 8 c = a = c = - b = 9-9 = b 55 = b C -, - = C(-, -) c = c = 9 x + 55 + y + =
Find an equation for the hyperbola described. ) center at (9, ); focus at (, ); vertex at (8, ) a = 9-8 c = 9 - + b = a = c = b = 5 a = c = x + 9 - y - 5 = Write the equation in standard form. Identify the conic section. ) x + y - x + 0y + = 0 x - x + y + 0y = - x - 8x + + y + 0y + 5 = - + + 00 x - + y + 5 = 8 x - + y + 5 = Find a polar equation for the conic. A focus is at the pole. 5) e = ; directrix is perpendicular to the polar axis to the right of the pole () + cos = 8 + cos Solve the problem. ) For the point (5, ), find other polar coordinates (r, ) of the point for which: (a) r > 0, - < 0 (b) r < 0, 0 < (c) r > 0 < a) (5, b) (-5, c) (5, - ) = (5, - + ) = (5, + + ) = (5, + ) = (5, - 9 ) )= (5, 7 ) )= (5, )
Graph the equation as accurately as possible. If it is a parabola, name and graph the vertex, focus and the directrix. Give the equation of the directrix. If it is an ellipse or hyperbola, then name and graph the verticies, foci, and center. Write the equations for the asymptotes and graph them. (0 POINTS EACH) 7) (x + ) - (y - 5) = x + - y - 5 = a = b = c = + = 7 C(-, 5) a = ± b = ± c = ± 7 V(- ±, 5) V(, 5) V(-9, 5) F(- ± 7, 5) asymptotes: y - 5 = ± x +
8) x - 8x = -8y - x - 8x = -8y - x - 8x + = -8y - + x - = -8y - 8 x - = -8(y + ) - 8 = p -8 = p - = p - 8 x - = y + directrix: y = - + = - - 8 x - - = y V(, -) F(, - - )= (, -8)
Plot the point given in polar coordinates. 9) (5, 5 ) The rectangular coordinates of a point are given. Find polar coordinates for the point. 0 < 0) (, -) + - = tan- - + = tan- - = - - + = - + =, The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, ). ) x + y = rcos + rsin = rcos + rsin = r cos + sin = Convert the polar equation to a rectangular equation set equal to zero. sec ) sec + Identify the shape ( points extra credit) cos cos + = + cos e = hyperbola 5
Identify the symmetry of the equation. Label each test. ) - sin = Polar Axis Pole - sin - - sin (-) - - sin - sin [ - ] sin () sin - [sin()cos() - cos()sin()] maybe maybe - [-sin ] sin maybe Determine the polar equation for the graph. ) x = - rcos = - Identify and graph the conic section that is represented by the equation below. Describe the location of the directrix. Find and graph the polar coordinates of the vertex(vertices) and the center. (0 POINTS)
5) - sin - sin = - sin e = ellipse = p = p The directrix i s parallel to the polar axis and units below the pole. - sin - - sin - - = V(, ) a = - - = - - Graph the polar equation. Make a table. = = = 8 V, C - 8, =, 7
) 5sin () Identify the shape ( points extra credit) rose with three petals = Polar Axis Pole 5 sin - 5 sin (-) - 5 sin 5 sin [ - ] -5 sin () -5 sin 5 [sin()cos() - cos()sin()] maybe maybe 5 [-(-sin )] 5 sin yes 5sin 0 0 0 5 5 0 0-5 5 0 5 8
BONUS. Solve. (0 POINTS) 7) A bridge is built in the shape of a semielliptical arch. It has a span of 0 feet. The height of the arch 0 feet from the center is to be 0 feet. Find the height of the arch 0 feet from its center. Round the answer to the nearest hundredth, if necessary. 9