# Graph the plane curve defined by the parametric equations: x = t^2 + 1, y = 3t, -2 leq t leq 2.

Question
Graph the plane curve defined by the parametric equations: $$x = t^2 + 1, y = 3t, -2 \leq t \leq 2$$.

2021-03-10

### Relevant Questions

Graph the plane curve defined by the parametric equations $$x = 125 cos t and y = 125 sin t$$
Determine the area of the region below the parametric curve given by the set of parametric equations. For each problem you may assume that each curve traces out exactly once from right to left for the given range of t. For these problems you should only use the given parametric equations to determine the answer. 1.$$x = t^2 + 5t - 1 y = 40 - t^2 -2 \leq t \leq 5$$ 2.$$x = 3cos^2 (t) — sin^2 (t) y = 6 + cos(t) -\frac{\pi}{2} \neq t \leq 0$$ 3.$$x = e^{\frac{1}{4} t} —2 y = 4 + e^{\frac{1}{4 t}} — e^{\frac{1}{4} t} - 6 \leq t \leq 1$$
Parametric equations and a value for the parameter t are given $$x = (60 cos 30^{\circ})t, y = 5 + (60 sin 30^{\circ})t - 16t2, t = 2$$. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.
Parametric to polar equations Find an equation of the following curve in polar coordinates and describe the curve. $$x = (1 + cos t) cos t, y = (1 + cos t) sin t, 0 \leq t \leq 2\pi$$
Working with parametric equations Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in x and y. b. Describe the curve and indicate the positive orientation. $$x=\ -7\ \cos\ 2t,\ y=\ -7\ \sin\ 2t,\ 0\ \leq\ t\ \leq\ \pi$$
Write parametric, equations for the given curve for the given definitions $$y =\ -4x\ +\ 1$$ a) $$x = t$$ b) $$x=\ \frac{t}{2}$$ c) $$x =\ -4t$$
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)$$
That parametric equations contain more information than just the shape of the curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $$x =\ \sin\ (t)\ and\ y =\ \cos\ (t)$$ where t represents time. We know that the shape of the path of the particle is a circle. a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.
Consider the helix represented investigation by the vector-valued function $$r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >$$. Solve for t in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s.