16. Parametric Equations & Polar Coordinates

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

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

$-2\le t\le 2$

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

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 

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = √t, y = 1 − √t, t ≥ 0

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

A circle centered at the origin with radius 4, generated counterclockwise

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

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

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

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

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x=√(t+1), y=√t, t ≥ 0

Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

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

c. Eliminate the parameter to obtain an equation in x and y.

d. Describe the curve.

x=2 t,y=3t−4;−10≤t≤10 

Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t