# 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.

 𝑡 0 0.5 1 1.5 2 𝑥 −2 −1 𝑎 1 2 𝑦 3 2.75 𝑏 𝑐 −1

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.

 𝑡 0 𝜋4 𝜋2 3𝜋4 𝜋 𝑥 2 √2 𝑎 −√2 −2 𝑦 0 −√22 𝑏 𝑐 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.

Q3:

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

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

Q6:

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

• A • B • C • D • E Q7:

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

• A • B • C • D • E Q8:

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

• A • B • C • D • E Q9:

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,