The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Conversion between Parametric and Rectangular Equations Mathematics • 12th Grade

In this worksheet, we will practice converting from the parametric form of an equation to its equivalent rectangular form and vice versa.


Use the fact that coshsinhπ‘₯βˆ’π‘₯=1 to find a parametrization of the part of the hyperbola π‘₯25βˆ’π‘¦81=1 that contains the point (βˆ’5,0).

  • Aπ‘₯=5𝑑,𝑦=9𝑑coshsinh
  • Bπ‘₯=βˆ’5𝑑,𝑦=βˆ’9𝑑coshsinh
  • Cπ‘₯=25𝑑,𝑦=81𝑑coshsinh
  • Dπ‘₯=βˆ’5𝑑,𝑦=9𝑑coshsinh
  • Eπ‘₯=5𝑑,𝑦=βˆ’9𝑑coshsinh


The first figure shows the graphs of cos2πœ‹π‘‘ and sin2πœ‹π‘‘ that parameterize the unit circle for 0≀𝑑≀1. What do the two functions graphed in the second figure parameterize?

  • AThe unit circle
  • BThe square on (0.5,0.5), (0.5,βˆ’0.5), (βˆ’0.5,βˆ’0.5), (βˆ’0.5,0.5)
  • CThe square on (0,0), (0,1), (1,1), (0,1)
  • DThe square on (1,0), (0,1), (βˆ’1,0), (0,βˆ’1)


Consider the points 𝐴=(1,1) and 𝐡=(5,4).

What is the length of 𝐴𝐡?

Find a parameterization of the segment 𝐴𝐡 over 0≀𝑑≀1.

  • Aπ‘₯=4(𝑑+1),𝑦=3(𝑑+1)
  • Bπ‘₯=𝑑+4,𝑦=𝑑+3
  • Cπ‘₯=4𝑑+1,𝑦=3𝑑+1
  • Dπ‘₯=3𝑑+1,𝑦=4𝑑+1
  • Eπ‘₯=3(𝑑+1),𝑦=4(𝑑+1)

Find 𝑓 and 𝑔 so that π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑) parameterizes 𝐴𝐡 over 0≀𝑑≀5.

  • A𝑓(𝑑)=4𝑑+1,𝑔(𝑑)=3𝑑+1
  • B𝑓(𝑑)=𝑑+4,𝑔(𝑑)=𝑑+3
  • C𝑓(𝑑)=3𝑑+1,𝑔(𝑑)=4𝑑+1
  • D𝑓(𝑑)=𝑑+45,𝑔(𝑑)=𝑑+35
  • E𝑓(𝑑)=45𝑑+1,𝑔(𝑑)=35𝑑+1

Using the functions above for 0≀𝑠≀5, what is the distance between the point (1,1) and the point (𝑓(𝑠),𝑔(𝑠))?

  • Aπ‘ βˆ’1
  • B𝑠+1
  • C𝑠
  • D𝑠+2
  • Eπ‘ βˆ’2

The parameterization of 𝐴𝐡 above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑) of 𝐴𝐢, where 𝐢=(13,6) and the parameter starts at 𝑑=0. Give the interval used.

  • Aπ‘₯=1213𝑑+1,𝑦=513𝑑+1 on 5≀𝑑≀13
  • Bπ‘₯=1213𝑑+1,𝑦=513𝑑+1 on 0≀𝑑≀13
  • Cπ‘₯=513𝑑+1,𝑦=1213𝑑+1 on 0≀𝑑≀13
  • Dπ‘₯=𝑑+1213,𝑦=513𝑑 on 0≀𝑑≀13
  • Eπ‘₯=513𝑑+1,𝑦=1213𝑑+1 on 5≀𝑑≀13


A particle moves along the curve given by the parametric equations π‘₯=2𝑑+1, 𝑦=βˆ’3𝑑+2 with βˆ’1√2≀𝑑≀1.

At which point is the particle located when 𝑑=1√2? Give your answer to one decimal place, if necessary.

  • A(2,1)
  • B(2,0.5)
  • C(1.5,0.5)
  • D(1,2)
  • E(0.5,2)

At which points is the particle located when 𝑑=0 and 𝑑=1? Give your answers to one decimal place, if necessary.

  • A(2,βˆ’3) and (1,3)
  • B(1,2) and (3,1)
  • C(2,1) and (βˆ’1,3)
  • D(1,2) and (3,βˆ’1)
  • E(1,2) and (3,5)

Find an equation for the line that the particle moves along in the form π‘Žπ‘₯+𝑏𝑦=𝑐.

  • A2π‘₯+3𝑦=7
  • B3π‘₯+2𝑦=7
  • C3π‘₯+2𝑦=13
  • D3π‘₯βˆ’2𝑦=7
  • E3π‘₯βˆ’2𝑦=βˆ’1

What is the smallest value of π‘₯ during the particle’s motion? When is it reached?

  • Aπ‘₯=1, at 𝑑=βˆ’1√2
  • Bπ‘₯=1, at 𝑑=0
  • Cπ‘₯=0.5, at 𝑑=βˆ’1√2
  • Dπ‘₯=0.5, at 𝑑=0
  • Eπ‘₯=βˆ’1, at 𝑑=1

Describe the motion from 𝑑=βˆ’1√2 to 𝑑=1 in terms of position on the line.

  • AThe particle starts at (0.5,2), goes left and upward to (2,1), and then goes back right and downward to (βˆ’1,3).
  • BThe particle starts at (2,0.5), goes left and upward to (1,2), and then goes back right and downward to (3,βˆ’1).
  • CThe particle starts at (2,0.5), goes right and downward to (1,2), and then goes back left and upward to (3,βˆ’1).
  • DThe particle starts at (1,2), goes left and upward to (2,1), and then goes back right and downward to (3,βˆ’1).
  • EThe particle starts at (0.5,2), goes right and downward to (2,1), and then goes back left and upward to (βˆ’1,3).

Give the parameters π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑) that describe the same motion but on an interval starting at 𝑑=0 instead of βˆ’1√2. In what interval does 𝑑 lie?

  • Aπ‘₯=ο€Ώπ‘‘βˆ’1√2+1, 𝑦=ο€Ώπ‘‘βˆ’1√2+1, the interval 0β‰€π‘‘β‰€βˆš2+1√2
  • Bπ‘₯=2𝑑+1√2ο‹βˆ’1, 𝑦=βˆ’3𝑑+1√2ο‹βˆ’1, the interval 0β‰€π‘‘β‰€βˆš2+1√2
  • Cπ‘₯=2𝑑+1√2+1, 𝑦=βˆ’3𝑑+1√2+1, the interval 0β‰€π‘‘β‰€βˆš2+1√2
  • Dπ‘₯=𝑑+1√2+1, 𝑦=𝑑+1√2+1, the interval 0β‰€π‘‘β‰€βˆš2+1√2
  • Eπ‘₯=2ο€Ώπ‘‘βˆ’1√2+1, 𝑦=βˆ’3ο€Ώπ‘‘βˆ’1√2+2, the interval 0β‰€π‘‘β‰€βˆš2+1√2


Convert the parametric equations π‘₯=𝑑+2 and 𝑦=3π‘‘βˆ’1 to rectangular form.

  • Aπ‘₯=𝑦+2
  • Bπ‘₯=𝑦+13+2
  • Cπ‘₯=𝑦+13+2
  • Dπ‘₯=3π‘¦βˆ’1
  • Eπ‘₯=𝑦+13


Convert the parametric equations π‘₯=2𝑑+1 and 𝑦=π‘‘βˆ’4 to the rectangular form.

  • A𝑦=π‘₯βˆ’92
  • B𝑦=2π‘₯+1
  • C𝑦=π‘₯+92
  • D𝑦=π‘₯βˆ’4
  • E𝑦=π‘₯βˆ’12


Convert the rectangular equation π‘₯+𝑦=25 to parametric form.

  • Aπ‘₯=𝑑,𝑦=25𝑑sincos
  • Bπ‘₯=5𝑑,𝑦=5𝑑cossin
  • Cπ‘₯=5𝑑,𝑦=5𝑑cossin
  • Dπ‘₯=𝑑,𝑦=𝑑sincos
  • Eπ‘₯=25𝑑,𝑦=25𝑑sincos


Convert the parametric equations π‘₯=3𝑑cos and 𝑦=3𝑑sin to rectangular form.

  • Aπ‘₯βˆ’π‘¦=9
  • B𝑦=3π‘₯sin
  • Cπ‘₯+𝑦=3
  • Dπ‘₯+𝑦=9
  • Eπ‘₯=3𝑦cos


Convert the rectangular equation π‘₯βˆ’π‘¦=9 to parametric form.

  • Aπ‘₯=3𝑑,𝑦=3𝑑coshsinh
  • Bπ‘₯=9𝑑,𝑦=9𝑑coshsinh
  • Cπ‘₯=3𝑑,𝑦=3𝑑cossin
  • Dπ‘₯=9𝑑,𝑦=9𝑑sincos
  • Eπ‘₯=3𝑑,𝑦=3𝑑sinhcosh


Consider the points 𝐴=(βˆ’1,1) and 𝐡=(4,2). Parameterize the segment 𝐴𝐡, where 0≀𝑑≀1.

  • Aπ‘₯=1βˆ’π‘‘,𝑦=5π‘‘βˆ’1
  • Bπ‘₯=𝑑+1,𝑦=5π‘‘βˆ’1
  • Cπ‘₯=5𝑑+1,𝑦=1βˆ’π‘‘
  • Dπ‘₯=5𝑑+1,𝑦=1+𝑑
  • Eπ‘₯=5π‘‘βˆ’1,𝑦=𝑑+1

This lesson includes 6 additional questions and 29 additional question variations for subscribers.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.