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 𝑥=2𝑡2 and 𝑦=3𝑡, where 0𝑡2.

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 𝑎,𝑏, and 𝑐.

  • A 𝑎 = 0 , 𝑏 = 2 , 𝑐 = 1 . 2 5
  • B 𝑎 = 0 , 𝑏 = 1 , 𝑐 = 0
  • C 𝑎 = 0 , 𝑏 = 2 , 𝑐 = 0 . 7 5
  • D 𝑎 = 2 , 𝑏 = 2 , 𝑐 = 0 . 7 5

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 𝑥=2𝑡cos and 𝑦=𝑡sin, where 0𝑡𝜋.

He has started to complete a table of values.

𝑡 0 𝜋 4 𝜋 2 3 𝜋 4 𝜋
𝑥 2 2 𝑎 2 2
𝑦 0 2 2 𝑏 𝑐 0

Work out the values of 𝑎,𝑏, and 𝑐.

  • A 𝑎 = 0 , 𝑏 = 1 , 𝑐 = 2 2
  • B 𝑎 = 2 , 𝑏 = 0 . 0 2 7 , 𝑐 = 0 . 0 4 1
  • C 𝑎 = 0 , 𝑏 = 1 , 𝑐 = 2 2
  • D 𝑎 = 2 , 𝑏 = 1 , 𝑐 = 2 2

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 𝑡 to be greater than 𝜋.
  • BDavid could extend his table of values; for example, he could increase 𝑡 by increments of 𝜋8 rather than 𝜋4.
  • CDavid could extend the value of 𝑡 to be lower than 0.

Q3:

A particle following the parameterization 𝑥=(2𝜋𝑡)cos, 𝑦=(2𝜋𝑡)sin of the unit circle starts at (1,0) and moves counterclockwise. At what values of 0𝑡4 is the particle at (0,1)? Give exact values.

  • A 1 4 , 5 4
  • B 1 4 , 3 4 , 5 4 , 7 4 , 9 4 , 1 1 4 , 1 3 4 , 1 5 4
  • C 1 4 , 3 4
  • D 1 4 , 3 4 , 5 4 , 7 4
  • E 1 4 , 5 4 , 9 4 , 1 3 4

Q4:

A particle following the parameterization 𝑥=2𝜋𝑡cos, 𝑦=2𝜋𝑡sin of the unit circle starts at (1,0) and moves counterclockwise. At what values of 𝑡0 is the particle at (0,1)? Give exact values.

  • A 𝑡 = 1 4 + 𝑛 , where 𝑛 is a nonnegative integer
  • B 𝑡 = 1 4 + 𝑛 , where 𝑛 is a nonnegative integer
  • C 𝑡 = 1 4 + 2 𝑛 , where 𝑛 is a nonnegative integer
  • D 𝑡 = 1 4 + 𝑛 , where 𝑛 is a nonnegative odd number
  • E 𝑡 = 1 4 + 𝑛 , where 𝑛 is a nonnegative integer

Q5:

A particle following the parameterization 𝑥=(𝑡)cos, 𝑦=(𝑡)sin of the unit circle starts at (1,0) and moves counterclockwise for 𝑡0. For what values of 𝑡 is the particle below the 𝑥-axis? Give exact values.

  • A 2 𝑛 𝜋 < 𝑡 < 𝜋 + 2 𝑛 𝜋 , with 𝑛=0,1,2,
  • BFor all real numbers
  • C 𝜋 2 + 2 𝑛 𝜋 < 𝑡 < 3 𝜋 2 + 2 𝑛 𝜋 , with 𝑛=0,1,2,
  • D 𝜋 + 2 𝑛 𝜋 < 𝑡 < 3 𝜋 2 + 2 𝑛 𝜋 , with 𝑛=0,1,2,
  • E 𝜋 + 2 𝑛 𝜋 < 𝑡 < 2 𝜋 + 2 𝑛 𝜋 , with 𝑛=0,1,2,

Q6:

Consider the parametric equations 𝑥(𝑡)=2𝑡sin and 𝑦(𝑡)=3𝑡cos, where 0<𝑡<3𝜋. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q7:

Consider the parametric equations 𝑥(𝑡)=𝑡1 and 𝑦(𝑡)=𝑡, where 2<𝑡<1. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q8:

Consider the parametric equations 𝑥(𝑡)=𝑡2 and 𝑦(𝑡)=3𝑡+1, where 2<𝑡<2. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q9:

Consider the parametric equations 𝑥(𝑡)=𝑡+2 and 𝑦(𝑡)=3𝑡1, where 2<𝑡<1. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q10:

Consider the parametric equations 𝑥(𝑡)=𝑒 and 𝑦(𝑡)=𝑒+1, where 4<𝑡<0.5. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q11:

Consider the parametric equations 𝑥(𝑡)=2𝑡 and 𝑦(𝑡)=23𝑡, where 0.5<𝑡<3. Which of the following is the sketch of the given equations?

  • A
  • B