Lesson: Conversion between Parametric and Rectangular Equations AP Calculus BC • Precalculus • Sullivan • Tenth Edition

Mathematics

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

Video

12:31

Sample Question Videos

  • 01:05
  • 01:51
  • 01:17

Worksheet: 15 Questions • 6 Videos

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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