Curves defined by parametric curves

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Discussion Overview

The discussion revolves around finding Cartesian equations from given parametric equations. Participants explore the relationships between the parameters and the resulting Cartesian forms, addressing both the mathematical transformations and the implications of parameter restrictions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests eliminating the parameter to find Cartesian equations for the curves defined by the parametric equations $x=\sin\frac{1}{2} \theta$ and $y=\cos\frac{1}{2} \theta$, as well as $x=e^t-1$ and $y=e^{2t}$.
  • Another participant hints at using the Pythagorean identity and suggests squaring the first equation to aid in the transformation.
  • There is a proposal that the second parametric equation could be expressed as $y=x^2+1$, which is later challenged and refined.
  • One participant acknowledges a mistake in disregarding the sign and proposes that the second equation can be rewritten as $y=(x+1)^2$ after manipulating $e^t=x+1$.
  • Another participant questions whether the restriction on $\theta$ needs to be considered beyond just graphing the equation.
  • There is a suggestion that the equation $x^2+y^2=1$ could be modified to include the condition $y \ge 0$ to account for the original parameter restrictions.
  • A later reply clarifies that the equation can be expressed as $y=\sqrt{1-x^2}$, which inherently satisfies the condition $0 \le y$.

Areas of Agreement / Disagreement

Participants express differing views on how to handle the restrictions imposed by the parameters, particularly regarding whether these restrictions should be explicitly included in the Cartesian equations. The discussion remains unresolved on the best approach to incorporate these conditions.

Contextual Notes

Participants do not reach a consensus on the necessity of including parameter restrictions in the final Cartesian equations, leading to varying interpretations of the equations derived from the parametric forms.

ineedhelpnow
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eliminate the parameter to find a cartesian equation of the curve.

$x=sin\frac{1}{2} \theta$
$y=cos\frac{1}{2} \theta$
$-\pi \le \theta \le \pi$

$x=e^t-1$
$y=e^{2t}$
 
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1.) Hint: Pythagorean identity.

2.) Hint: Square the first equation...
 
is the second one $y=x^2+1$
 
ineedhelpnow said:
is the second one $y=x^2+1$

No, the first equation may be written as:

$$e^t=x+1$$

Square:

$$e^{2t}=(x+1)^2$$

$$y=(x+1)^2$$
 
oh i see. i just disregarded the sign for some reason. and that second one would be $x^2+y^2=1$, right? square both sides and then use identity which simplifies one side of the equation to 1
 
ineedhelpnow said:
oh i see. i just disregarded the sign for some reason. and that second one would be $x^2+y^2=1$, right? square both sides and then use identity which simplifies one side of the equation to 1

Yes...but do you need to account for the restriction on $\theta$?
 
i don't know. only when graphing it?
 
ineedhelpnow said:
only when graphing it?

Well, since the graph represents the set of points that satisfy the given equation, it's not only the graph you need worry about. Does every point on the circle satisfy all of the original conditions? If not, can we write the equation is such a way that all initial conditions are satisfied?
 
isnt it just $x^2+y^2=1$ $y \ge 0$
 
  • #10
ineedhelpnow said:
isnt it just $x^2+y^2=1$ $y \ge 0$

Yes, or you could just write:

$$y=\sqrt{1-x^2}$$

This describes the curve and ensures $0\le y$ all in one statement.
 
  • #11
oh ok. i see what you mean now.
 

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