# Lesson Worksheet: Parametric Equations and Curves in Two Dimensions Mathematics • 12th Grade

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

Q1:

Noah 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:

Jackson wants to graph the parametric equations and , where .

He has started to complete a table of values.

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

Work out the values of , , and .

• A, , and
• B, , and
• C, , and
• D, , and

When Jackson 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?

• AJackson could extend the value of to be greater than .
• BJackson could refine his table of values; for example, he could increase by increments of rather than .
• CJackson 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:

The coordinates of a particle at time are described by the parameterization , . The particle starts at and follows the unit circle in a counterclockwise direction. At what values of are the coordinates of the particle ? 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 This lesson includes 7 additional questions and 36 additional question variations for subscribers.