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(a) What does the equation y = x2 represent as a curve in R2? circle parabola hyperbola ellipse line (b) What does it represent as a surface in R3? parabolic cylinder ellipsoid cone hyperboloid elliptic paraboloid (c) What does the equation z = y2 represent? parabolic cylinder elliptic paraboloid cone ellipsoid hyperboloid

(a) What does the equation y = x2 represent as a curve in R2? circle parabola hyperbola ellipse line (b) What does it represent as a surface in R3? parabolic cylinder ellipsoid cone hyperboloid elliptic paraboloid (c) What does the equation z = y2 represent? parabolic cylinder elliptic paraboloid cone ellipsoid hyperboloid

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(a) What does the equation y = x 2 represent as a curve in R 2 ? circle parabola hyperbola ellipse line (b) What does it represent as a surface in R 3 ? parabolic cylinder ellipsoid cone hyperboloid elliptic paraboloid (c) What does the equation z = y 2 represent? parabolic cylinder elliptic paraboloid cone ellipsoid hyperboloid

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(a) Find and identify the traces of the quadric surface x2 + y2 − z2 = 64 given the plane. x = k Find the trace. Identify the trace. circle ellipse hyperbola parabola y = k Find the trace. Identify the trace. circle ellipse hyperbola parabola z = k Find the trace. Identify the trace. circle ellipse hyperbola parabola Describe the surface from one of the graphs in the table. ellipsoid elliptic paraboloid hyperbolic paraboloid cone hyperboloid of one sheet hyperboloid of two sheets (b) If we change the equation in part (a) to x2 − y2 + z2 = 64, how is the graph affected? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction. (c) What if we change the equation in part (a) to x2 + y2 + 2y − z2 = 63? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction.(a) Find and identify the traces of the quadric surface x2 + y2 − z2 = 64 given the plane. x = k Find the trace. Identify the trace. circle ellipse hyperbola parabola y = k Find the trace. Identify the trace. circle ellipse hyperbola parabola z = k Find the trace. Identify the trace. circle ellipse hyperbola parabola Describe the surface from one of the graphs in the table. ellipsoid elliptic paraboloid hyperbolic paraboloid cone hyperboloid of one sheet hyperboloid of two sheets (b) If we change the equation in part (a) to x2 − y2 + z2 = 64, how is the graph affected? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction. (c) What if we change the equation in part (a) to x2 + y2 + 2y − z2 = 63? The graph is rotated so that its axis is the x-axis. The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axis The graph is shifted one unit in the negative y-direction. The graph is shifted one unit in the positive y-direction.
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Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) x = y2 + 3z2 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = −3 using y and z.) (Write an equation for the cross section at x = 0 using y and z.) (Write an equation for the cross section at x = 3 using y and z.) Identify the surface. elliptic cone parabolic cylinder hyperbolic paraboloid elliptic paraboloid elliptic cylinder hyperboloid of two sheets ellipsoid hyperboloid of one sheetUse traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) x = y2 + 3z2 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = −3 using y and z.) (Write an equation for the cross section at x = 0 using y and z.) (Write an equation for the cross section at x = 3 using y and z.) Identify the surface. elliptic cone parabolic cylinder hyperbolic paraboloid elliptic paraboloid elliptic cylinder hyperboloid of two sheets ellipsoid hyperboloid of one sheet
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