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 π‘₯=2π‘‘βˆ’2 and 𝑦=3βˆ’π‘‘οŠ¨, where 0≀𝑑≀2.

He has started to complete a table of values.

𝑑00.511.52
π‘₯βˆ’2βˆ’1π‘Ž12
𝑦32.75π‘π‘βˆ’1

Find the values of π‘Ž,𝑏, and 𝑐.

  • Aπ‘Ž=0, 𝑏=2, 𝑐=1.25
  • Bπ‘Ž=0, 𝑏=1, 𝑐=0
  • Cπ‘Ž=0, 𝑏=2, 𝑐=0.75
  • Dπ‘Ž=2, 𝑏=2, 𝑐=0.75

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 π‘₯=2𝑑cos and 𝑦=βˆ’π‘‘sin, where 0β‰€π‘‘β‰€πœ‹.

He has started to complete a table of values.

𝑑0πœ‹4πœ‹23πœ‹4πœ‹
π‘₯2√2π‘Žβˆ’βˆš2βˆ’2
𝑦0βˆ’βˆš22𝑏𝑐0

Work out the values of π‘Ž,𝑏, and 𝑐.

  • Aπ‘Ž=0, 𝑏=1, 𝑐=√22
  • Bπ‘Ž=2, 𝑏=βˆ’0.027, 𝑐=βˆ’0.041
  • Cπ‘Ž=0, 𝑏=βˆ’1, 𝑐=βˆ’βˆš22
  • Dπ‘Ž=2, 𝑏=βˆ’1, 𝑐=βˆ’βˆš22

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 𝑑 to be greater than πœ‹.
  • BDavid could extend his table of values; for example, he could increase 𝑑 by increments of πœ‹8 rather than πœ‹4.
  • CDavid could extend the value of 𝑑 to be lower than 0.

Q3:

A particle following the parameterization π‘₯=(2πœ‹π‘‘)cos, 𝑦=(2πœ‹π‘‘)sin of the unit circle starts at (1,0) and moves counterclockwise. At what values of 0≀𝑑≀4 is the particle at (0,1)? Give exact values.

  • A14,34,54,74,94,114,134,154
  • B14,34,54,74
  • C14,54
  • D14,54,94,134
  • E14,34

Q4:

A particle following the parameterization π‘₯=ο€Ή2πœ‹π‘‘ο…cos, 𝑦=ο€Ή2πœ‹π‘‘ο…sin of the unit circle starts at (1,0) and moves counterclockwise. At what values of 𝑑β‰₯0 is the particle at (0,1)? Give exact values.

  • A𝑑=ο„ž14+𝑛, where 𝑛 is a nonnegative integer
  • B𝑑=14+𝑛, where 𝑛 is a nonnegative integer
  • C𝑑=ο„ž14+2𝑛, where 𝑛 is a nonnegative integer
  • D𝑑=ο„ž14+𝑛, where 𝑛 is a nonnegative odd number
  • E𝑑=ο„ž14+𝑛, where 𝑛 is a nonnegative integer

Q5:

A particle following the parameterization π‘₯=(𝑑)cos, 𝑦=(𝑑)sin of the unit circle starts at (1,0) and moves counterclockwise for 𝑑β‰₯0. For what values of 𝑑 is the particle below the π‘₯-axis? Give exact values.

  • A2π‘›πœ‹<𝑑<πœ‹+2π‘›πœ‹, with 𝑛=0,1,2,…
  • BFor all real numbers
  • Cπœ‹2+2π‘›πœ‹<𝑑<3πœ‹2+2π‘›πœ‹, with 𝑛=0,1,2,…
  • Dπœ‹+2π‘›πœ‹<𝑑<3πœ‹2+2π‘›πœ‹, with 𝑛=0,1,2,…
  • Eπœ‹+2π‘›πœ‹<𝑑<2πœ‹+2π‘›πœ‹, with 𝑛=0,1,2,…

Q6:

Consider the parametric equations π‘₯(𝑑)=2𝑑sin and 𝑦(𝑑)=3𝑑cos, where 0<𝑑<3πœ‹. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q7:

Consider the parametric equations π‘₯(𝑑)=π‘‘βˆ’1 and 𝑦(𝑑)=𝑑, where βˆ’2<𝑑<1. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q8:

Consider the parametric equations π‘₯(𝑑)=π‘‘βˆ’2 and 𝑦(𝑑)=3𝑑+1, where βˆ’2<𝑑<2. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q9:

Consider the parametric equations π‘₯(𝑑)=𝑑+2 and 𝑦(𝑑)=3π‘‘βˆ’1, where βˆ’2<𝑑<1. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q10:

Consider the parametric equations π‘₯(𝑑)=𝑒 and 𝑦(𝑑)=𝑒+1, where βˆ’4<𝑑<0.5. Which of the following is the sketch of the given equations?

  • A
  • B
  • C
  • D
  • E

Q11:

Consider the parametric equations π‘₯(𝑑)=2βˆ’π‘‘ and 𝑦(𝑑)=2βˆ’3𝑑, where 0.5<𝑑<3. 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 𝑓 and 𝑔 respectively. Describe the curve parameterized by π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑).

  • AThe square on vertices 𝐴(1,1), 𝐡(3,3), 𝐢(5,3), and 𝐷(7,1) traced as 𝐴𝐡𝐢𝐷𝐴
  • BThe square on vertices 𝐴(1,1), 𝐡(3,1), 𝐢(3,3), and 𝐷(1,3) traced as 𝐴𝐡𝐢𝐷𝐴
  • CThe square on vertices 𝐴(1,1), 𝐡(1,3), 𝐢(3,5), and 𝐷(5,1) traced as 𝐴𝐡𝐢𝐷𝐴
  • DThe square on vertices 𝐴(1,1), 𝐡(3,3), 𝐢(3,5), and 𝐷(1,7) traced as 𝐴𝐡𝐢𝐷𝐴
  • EThe square on vertices 𝐴(1,1), 𝐡(1,3), 𝐢(3,3), and 𝐷(3,1) traced as 𝐴𝐡𝐢𝐷𝐴

Q13:

By sketching the circle with parametric equations π‘₯=𝑑+1cos and 𝑦=π‘‘βˆ’2sin, where βˆ’πœ‹β‰€π‘‘β‰€πœ‹, or otherwise, determine its center and radius.

  • ACenter:radius:(1,βˆ’2),2
  • BCenter:radius:(βˆ’2,1),2
  • CCenter:radius:(βˆ’1,2),2
  • DCenter:radius:(βˆ’1,2),1
  • ECenter:radius:(1,βˆ’2),1

Q14:

Daniel wants to graph the parametric curve defined by the equations π‘₯=𝑑+1 and 𝑦=5π‘‘βˆ’1 for βˆ’2≀𝑑≀2. Determine the coordinates of the point on the curve where 𝑑=1.

  • A(1,βˆ’1)
  • B(4,2)
  • C(0,βˆ’6)
  • D(βˆ’6,0)
  • E(2,4)

Q15:

Determine the values of π‘Ž and 𝑏 by finding the dimensions of the smallest possible rectangle, in the orientation shown, that would contain the parametric curve defined by the equations π‘₯=2(5𝑑)cos and 𝑦=(4𝑑)sin, where βˆ’πœ‹β‰€π‘‘β‰€πœ‹.

  • Aπ‘Ž=2, 𝑏=4
  • Bπ‘Ž=8, 𝑏=4
  • Cπ‘Ž=1, 𝑏=2
  • Dπ‘Ž=2, 𝑏=1
  • Eπ‘Ž=4, 𝑏=2

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