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

In this lesson, we will learn how to convert from the parametric form of an equation to its equivalent rectangular form and vice versa.

Lesson Plan

Students will be able to

  • convert a given pair of parametric equations into rectangular form by elimination without domain restriction,
  • convert a given pair of parametric equations into rectangular form by elimination while considering the domain,
  • convert a given pair of parametric equations into rectangular form by applying an identity,
  • convert a rectangular equation into parametric form by using a given expression for a parameter,
  • convert a given pair of parametric equations into rectangular form and then sketch its graph.

Lesson Video

Video Thumbnail
12:31

Lesson Playlist

Lesson Worksheet

Q1:

Use the fact that coshsinh𝑥−𝑥=1 to find a parametrization of the part of the hyperbola 𝑥25−𝑦81=1 that contains the point (−5,0).

Q2:

The first figure shows the graphs of cos2𝜋𝑡 and sin2𝜋𝑡 that parameterize the unit circle for 0≤𝑡≤1. What do the two functions graphed in the second figure parameterize?

Q3:

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

What is the length of 𝐴𝐵?

Find a parameterization of the segment 𝐴𝐵 over 0≤𝑡≤1.

Find 𝑓 and 𝑔 so that 𝑥=𝑓(𝑡), 𝑦=𝑔(𝑡) parameterizes 𝐴𝐵 over 0≤𝑡≤5.

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

The parameterization of 𝐴𝐵 above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization 𝑥=𝑓(𝑡), 𝑦=𝑔(𝑡) of 𝐴𝐶, where 𝐶=(13,6) and the parameter starts at 𝑡=0. Give the interval used.

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