16. Parametric Equations & Polar Coordinates

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=-t+1$; $y\left(t\right)=t^2$

$-2\le t\le 2$

Graph the plane curve formed by the parametric equations and indicate its orientation.

$x\left(t\right)=2t-1$; $y\left(t\right)=2\sqrt{t}$

$t\ge 0$

Eliminate the parameter to rewrite the following as a rectangular equation.

$x\left(t\right)=2t-1$

$y\left(t\right)=t^5-2$

First eliminate the parameter, then graph the plane curve of the parametric equations.

$x\left(t\right)=2+\cos t$, $y\left(t\right)=-1+\sin t$; $0\le t\le2\pi$

Write parametric equations for the rectangular equation below.

$x^2+y^2=25$

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=2 t,y=3t−4;−10≤d≤10 

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=−t+6, y=3t−3; −5≤t≤5 

15–30. 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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

15–30. 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 = √t + 4, y = 3√t; 0 ≤ t ≤ 16

15–30. 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 = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π

15–30. 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 = cos t, y = sin² t; 0 ≤ t ≤ π

15–30. 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 = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

15–30. 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 = r − 1, y = r³; −4 ≤ r ≤ 4

15–30. 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 = 8 + 2t, y = 1; −∞ < t < ∞

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=2 sin 8t, y=2 cos 8t