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Wize University Calculus 2 Textbook > Parametric & Polar Curves

Parametric Equations

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Parametric Curves

A Cartesian equation is give by y(x)=... x ...y\left(x\right)=...\ x\ ...
A curve/graph can be defiend by points written in the form (x, y(x))\left(x,\ y\left(x\right)\right)-- called rectangular coordinates

Parametric Equations & Curves
  • Parametric equations x(t), y(t)x(t),\ y(t) depend on a parameter tt
  • Parametric curves are defined by points written in the form (x, y)=(x(t), y(t))\left(x,\ y\right)=\left(x\left(t\right),\ y\left(t\right)\right)
  • If t∈[a,b]t\in\left[a,b\right], then (x(a), y(a))\left(x\left(a\right),\ y\left(a\right)\right) is called the initial point and (x(b), y(b))\left(x\left(b\right),\ y\left(b\right)\right) is called the terminal point.

Strategies for Graphing in Parametric Curves
  1. If possible, rearrange to solve for tt in terms of xxand substitute this into the yyequation
  2. Try to plot a few points for nice values of tt within the defined interval [a,b][a, b]


Example
Sketch the curve of the given parametric equations x=t−4,    y=3+t2,    t∈R.x=t-4,\ \ \ \ y=3+t^2,\ \ \ \ t\in\mathbb{R}.
Method 1
Let us rearrange to solve for y in terms of x. x.\
t=x+4,t=x+4,so y=3+(x+4)2 y = 3 + (x + 4)^2.
Also notice that there are no restrictions on the x value (when t∈R  →  x∈Rt\in\mathbb{R}\ \ \to\ \ x\in\mathbb{R}
So we can recognize this curve as a parabola with vertex (-4, 3).

Method 2
We can pick some values for t and out a few (x,y) points.
t=0:  x=−4, y=3t=0:\ \ x=-4,\ y=3
t=1:  x=−3, y=4t=1:\ \ x=-3,\ y=4
t=−1:  x=−5, y=4t=-1:\ \ x=-5,\ y=4
Continue to plot a few more points and you will see that this graph is a parabola with vertex (-4, 3).

Practice Question

Write the Cartesian equation that corresponds to the parametric equations {x=cos⁡t, y=cos⁡2t: t∈R.}\left\{x=\cos t,\ y=\cos^2t:\ t\in\mathbb{R}.\right\}, and state any restrictions on the variables.

Practice Question

Sketch the parametric curve x=2cos⁡3t,  y=2sin⁡3tx=2\cos3t,\ \ y=2\sin3t, where t∈[0,2π]t\in\left[0,2\pi\right].

Practice Question

Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.




A.
ii.
B.
iii.
C.
i.
a.
b.
c.
Extra Practice
Practice: Parametric Curve
Write the Cartesian equation that corresponds to the parametric equatoins {x=cos⁡t, y=cos⁡2t: t∈R.}\left\{x=\cos t,\ y=\cos^2t:\ t\in\mathbb{R}.\right\}, and state any restrictions on the variables.
Practice: Parametric Curve
Practice Question
Write the Cartesian equation that corresponds to the parametric equatoins {x=cos⁡t, y=cos⁡2t: t∈R.}\left\{x=\cos t,\ y=\cos^2t:\ t\in\mathbb{R}.\right\}, and state any restrictions on the variables.
Practice: Parametric Curves Graphs
Practice Question
Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.
Practice: Parametric Curves Graphs
Practice Question
Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.
Practice: Parametric Curves Graphs
Practice Question
Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.
Practice: Parametric Curve
Practice Question
Write the Cartesian equation that corresponds to the parametric equatoins {x=cos⁡t, y=cos⁡2t: t∈R.}\left\{x=\cos t,\ y=\cos^2t:\ t\in\mathbb{R}.\right\}, and state any restrictions on the variables.
Practice: Parametric Curves Graphs
Practice Question
Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.
Practice: Parametric Curves Graphs
Practice Question
Given the graphs a. to c. of x(t), y(t)x\left(t\right),\ y\left(t\right), match them with the appropriate parametric curves i. to iii. of {x(t), y(t)}\left\{x\left(t\right),\ y\left(t\right)\right\}.
Practice: Parametric Curve
Practice Question
Write the Cartesian equation that corresponds to the parametric equations {x=cos⁡t, y=cos⁡2t: t∈R.}\left\{x=\cos t,\ y=\cos^2t:\ t\in\mathbb{R}.\right\}, and state any restrictions on the variables.