# Worksheet: Conversion between Parametric and Rectangular Equations

In this worksheet, we will practice converting from the parametric form of an equation to its equivalent rectangular form and vice versa.

**Q1: **

Use the fact that to find a parametrization of the part of the hyperbola that contains the point .

- A
- B
- C
- D
- E

**Q2: **

The first figure shows the graphs of and , which parameterize the unit circle for . What do the two functions graphed in the second figure parameterize?

- Athe unit circle
- Bthe square on , , ,
- Cthe square on , , ,
- Dthe square on , , ,

**Q3: **

Consider the points and .

What is the length of ?

Find a parameterization of the segment over .

- A
- B
- C
- D
- E

Find and so that , parameterizes over .

- A
- B
- C
- D
- E

Using the functions above for , what is the distance between the point and the point ?

- A
- B
- C
- D
- E

The parameterization of above is an example of an
{*arc-length parameterization*} of a plane curve.
Find an arc-length parameterization , of
, where and the parameter starts at
. Give the interval used.

- A on
- B on
- C on
- D on
- E on

**Q4: **

A particle moves along the curve given by the parametric equations , with .

At which point is the particle located when ? Give your answer to one decimal place, if necessary.

- A
- B
- C
- D
- E

At which points is the particle located when and ? Give your answers to one decimal place, if necessary.

- A and
- B and
- C and
- D and
- E and

Find an equation for the line that the particle moves along in the form .

- A
- B
- C
- D
- E

What is the smallest value of during the particle’s motion? When is it reached?

- A , at
- B , at
- C , at
- D , at
- E , at

Describe the motion from to in terms of position on the line.

- AThe particle starts at , goes left and upward to , and then goes back right and downward to .
- BThe particle starts at , goes left and upward to , and then goes back right and downward to .
- CThe particle starts at , goes right and downward to , and then goes back left and upward to .
- DThe particle starts at , goes left and upward to , and then goes back right and downward to .
- EThe particle starts at , goes right and downward to , and then goes back left and upward to .

Give the parameters , that describe the same motion but on an interval starting at instead of . In what interval does lie?

- A , , the interval
- B , , the interval
- C , , the interval
- D , , the interval
- E , , the interval

**Q5: **

Convert the parametric equations and to rectangular form.

- A
- B
- C
- D
- E

**Q6: **

Convert the rectangular equation to parametric form.

- A
- B
- C
- D
- E

**Q7: **

Convert the parametric equations and to rectangular form.

- A
- B
- C
- D
- E

**Q8: **

Convert the rectangular equation to parametric form.

- A
- B
- C
- D
- E

**Q9: **

Consider the points and . Parameterize the segment , where .

- A
- B
- C
- D
- E

**Q11: **

Convert the rectangular equation to parametric form.

- A
- B
- C
- D
- E

**Q13: **

Convert the parametric equations and to rectangular form.

- A
- B
- C
- D
- E

**Q14: **

Convert the parametric equations and to rectangular form.

- A
- B
- C
- D
- E