Match the parametric curve to its description. Be careful: It is possible that the same description fits more than one parametric curve! (a) (t + 1,4t−2) (1) A portion of a parabola. (b) (t2,3t2) (2) A straight line traversed at constant speed. (c) (cost,2(cost)2 + cost) (3) A straight line traversed at variable speed.
(d) ((2t + 1)sint,(2t + 1)cost) (4) A circle. (e) (2t−1,5) (5) Doesn’t match any other description.
(a) (t + 1,4t−2) - (2) A straight line traversed at constant speed.
(b) (t2,3t2) - (1) A portion of a parabola.
(c) (cost,2(cost)2 + cost) - (5) Doesn’t match any other description.
(d) ((2t + 1)sint,(2t + 1)cost) - (4) A circle.
(e) (2t−1,5) - (3) A straight line traversed at variable speed.
To match each parametric curve to its description, we can examine the properties of the curves.
(a) (t + 1,4t-2): This curve is a straight line since both the x and y coordinates are linear functions of t.
(b) (t^2,3t^2): This curve is a portion of a parabola because both the x and y coordinates are quadratic functions of t.
(c) (cos(t),2(cos(t))^2 + cos(t)): This curve is a circle since both the x and y coordinates are trigonometric functions.
(d) ((2t + 1)sin(t),(2t + 1)cos(t)): This curve is a straight line traversed at variable speed since both the x and y coordinates involve a combination of trigonometric and linear functions.
(e) (2t - 1,5): This curve doesn't match any other description since both the x and y coordinates are linear functions of t, but the slope of the line is fixed and not dependent on t.
Therefore, the matching is:
(a) (t + 1,4t-2) -> (2) A straight line traversed at constant speed.
(b) (t^2,3t^2) -> (1) A portion of a parabola.
(c) (cos(t),2(cos(t))^2 + cos(t)) -> (4) A circle.
(d) ((2t + 1)sin(t),(2t + 1)cos(t)) -> (3) A straight line traversed at variable speed.
(e) (2t - 1,5) -> (5) Doesn't match any other description.