# Worksheet: Parametric Equations and Curves in Two Dimensions

In this worksheet, we will practice graphing curves given by a pair of parametric equations in 2D.

**Q1: **

Anthony wants to graph the parametric equations and , where .

He has started to complete a table of values.

0 | 0.5 | 1 | 1.5 | 2 | |

β2 | β1 | 1 | 2 | ||

3 | 2.75 | β1 |

Find the values of , and .

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

Use the table of values to determine which of the following graphs is correct.

- A
- B
- C
- D

**Q2: **

David wants to graph the parametric equations and , where .

He has started to complete a table of values.

0 | |||||

2 | |||||

0 | 0 |

Work out the values of , and .

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

When David plots the coordinates on a graph, he is not entirely sure about the shape of the curve. What is the one thing he could do to find out more about the shape of the curve?

- ADavid could extend the value of to be greater than .
- BDavid could extend his table of values; for example, he could increase by increments of rather than .
- CDavid could extend the value of to be lower than 0.

**Q4: **

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 integer
- D, where is a nonnegative odd number
- E, where is a nonnegative integer

**Q5: **

A particle following the parameterization , of the unit circle starts at and moves counterclockwise for . For what values of is the particle below the -axis? Give exact values.

- A, with
- BFor all real numbers
- C, with
- D, with
- E, with

**Q7: **

Consider the parametric equations and , where . Which of the following is the sketch of the given equations?

- A
- B
- C
- D
- E

**Q10: **

Consider the parametric equations and , where . Which of the following is the sketch of the given equations?

- A
- B
- C
- D
- E

**Q11: **

Consider the parametric equations and , where . Which of the following is the sketch of the given equations?

- A
- B
- C
- D
- E

**Q12: **

Figures (a) and (b) show the graphs of functions and respectively. Describe the curve parameterized by , .

- AThe square on vertices , , , and traced as
- BThe square on vertices , , , and traced as
- CThe square on vertices , , , and traced as
- DThe square on vertices , , , and traced as
- EThe square on vertices , , , and traced as

**Q13: **

By sketching the circle with parametric equations and , where , or otherwise, determine its center and radius.

- A
- B
- C
- D
- E

**Q14: **

Daniel wants to graph the parametric curve defined by the equations and for . Determine the coordinates of the point on the curve where .

- A
- B
- C
- D
- E

**Q15: **

Determine the values of and by finding the dimensions of the smallest possible rectangle, in the orientation shown, that would contain the parametric curve defined by the equations and , where .

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