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.

Well, I suppose if you do not see it as is, you can do x and y

for example in (a)
y = 4t-2
x = t+1
so
t = x-1
y=4(x-2) - 2
y = 4 x -10
dx/dt=dy/dt = 0 so constant speed

(d) If you do no see it is a circle, do the same thing again
y = (2t+1) cos t
y^2 = (2t+1)^2 cos^2 t
x^2 = (2t+1)^2 sin^2 t

x^2 + y^2 = (2t+1)^2
circle of expanding radius

I used a graphing site for parametric curve and (d) gives me a graph like a roller coaster. so it is a circle ?how about for b and e??

LOL e is a horizontal line, y = 5

b is a parabola, do what I suggested.

so what description(1-5)fits the given parametric curves(a-b)? cheers.

To match the parametric curves to their descriptions, we can analyze the equations of the curves and compare them to the descriptions provided.

(a) (t + 1, 4t - 2)
This is a linear equation where both x and y are linear functions of t. The x-coordinate increases by 1 as t increases, and the y-coordinate increases by 4 as t increases. This corresponds to a straight line.

(b) (t^2, 3t^2)
Both the x and y-coordinates are quadratic functions of t. This corresponds to a portion of a parabola.

(c) (cos t, 2(cos t)^2 + cos t)
The x-coordinate is given by the cosine of t, which implies it oscillates between -1 and 1. The y-coordinate is a function of the cosine of t, squared, and the cosine of t itself. This implies that as the x-coordinate oscillates, the y-coordinate also oscillates, but with an additional transformation. This does not match any of the descriptions.

(d) ((2t + 1)sin t, (2t + 1)cos t)
The x and y-coordinates are both sinusoidal functions of t, but with an added transformation of (2t + 1). This corresponds to a circle.

(e) (2t - 1, 5)
The x-coordinate is a linear function of t, increasing by 2 as t increases. The y-coordinate is a constant 5. This corresponds to a straight line traversed at constant speed.

Matching the descriptions to their corresponding parametric curves:
(a) - (2) A straight line traversed at constant speed.
(b) - (1) A portion of a parabola.
(c) - (5) Doesn’t match any other description.
(d) - (4) A circle.
(e) - (3) A straight line traversed at variable speed.