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)
(b) (t^2,3t^2)
(c) (cost,2(cost)^2 + cost)
(d) ((2t + 1)sint,(2t + 1)cost)
(e) (2t−1,5)
(1) A portion of a parabola.
(2) A straight line traversed at constant speed.
(3) A straight line traversed at variable speed.
(4) A circle.
(5) Doesn’t match any other description.
The parametric curve that I know the description is only (c) which is a (1) parabola. So how about the others?
visit wolframalpha.com and type in your functions, as in (d) below
http://www.wolframalpha.com/input/?i=parametric+plot+x%3D%282t+%2B+1%29sint,+y%3D%282t+%2B+1%29cost
So this means that (d) doesnt match any other description? because that's not a circle right?
What does (2) and (3) should look like in the graph?
What does (2)straight line traversed at constant speed and (3) A straight line traversed at variable speed should look like in the graph?
To determine the matches between the remaining parametric curves and their descriptions, we need to analyze the equations and characteristics of each curve.
(a) (t + 1, 4t - 2):
This is a linear equation in parametric form. The x-coordinate increases linearly with t, while the y-coordinate increases at four times the rate of t. Therefore, this represents a straight line traversed at a variable speed.
(b) (t^2, 3t^2):
This curve represents a quadratic equation in parametric form. Both the x and y coordinates increase quadratically with t, suggesting a parabolic shape. Therefore, this curve represents a portion of a parabola.
(c) (cos t, 2(cos t)^2 + cos t):
As mentioned before, this curve represents a parabola as its x-coordinate is given by the cosine of t, and the y-coordinate is determined by a quadratic function of the cosine of t. Therefore, this curve also matches the description of a portion of a parabola.
(d) ((2t + 1)sin t, (2t + 1)cos t):
This curve consists of trigonometric functions in parametric form. The x-coordinate is given by the sine of t multiplied by (2t + 1), and the y-coordinate is determined by the cosine of t multiplied by (2t + 1). These equations represent a curve that spirals outward continuously and does not meet any of the given descriptions.
(e) (2t - 1, 5):
This curve represents a linear equation in parametric form. The x-coordinate increases linearly with t, while the y-coordinate remains constant at 5. Therefore, this curve matches the description of a straight line traversed at a constant speed.
Based on the analysis above, the corresponding matches are as follows:
(a) - (3) A straight line traversed at variable speed.
(b) - (1) A portion of a parabola.
(c) - (1) A portion of a parabola.
(d) - (5) Doesn't match any other description.
(e) - (2) A straight line traversed at constant speed.