In this worksheet, we will practice finding the parametric equations of a plane curve.

**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: **

A particle following the parameterization , of the unit circle starts at and moves counterclockwise. At what values of is the particle at ? Give exact values.

- A
- B
- C
- D
- E

**Q3: **

The diagram shows a parabola that is symmetrical about the -axis and whose vertex is at the origin. It can be described by the parametric equations and , , where is a positive constant. The focus of the parabola is the point , and the directrix is the line with the equation .

Find a pair of parametric equations that describe the parabola whose focus is the point and whose directrix is the line . Include the parameter range.

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

**Q4: **

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 , , ,

**Q5: **

A particle following the parameterization , of the unit circle starts at and moves counterclockwise. At what values of is the particle at ? Give exact values.

- A , where is a nonnegative integer
- B , where is a nonnegative integer
- C , where is a nonnegative odd number
- D , where is a nonnegative integer
- E , where is a nonnegative integer

**Q6: **

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