apcalcParametric+Equations

Parametric Equations Emily Daughan







__Contents __ -Definition- -How to Graph- -Patterns- -Real-World Applications- -History-





= __Definition__ = =A pair of functions= =x=f(t) and y=g(t)= =which describe the x and y coordinates= =of the graph of some curve in a plane.=

=__How to Graph __= **Example: **Plot the system of parametric equations **//x=t^2// **and **//y=t^2+2t-3// **from t=-5 to t=5.

<span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">1) Make a table of values.


 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-add-space: auto; text-align: center;"><span style="color: #000000; display: block; font-family: 'Tahoma','sans-serif'; font-size: 16pt; text-align: center;">t || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-5 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-4 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-3 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-2 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-1 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">0 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">1 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">2 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">3 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">4 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">5 ||
 * <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">x=t^2 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">25 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">16 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">9 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">4 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">1 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">0 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">1 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">4 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">9 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">16 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">25 ||
 * <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">y=t^2+2t-3 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">12 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">5 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">0 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-3 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-4 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">-3 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">0 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">5 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">12 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 0pt;">21 || <span style="color: #000000; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">32 ||

<span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt;">2) Plot the points on the plane, with the values of t^2 serving as the x coordinates and the values of t^2+2t-3 serving as the y coordinates. Make sure that for each point, you used the same values of t to find x and y.

<span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt;">

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__<span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 22pt;">Patterns __ <span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">If the x-coordinate equation is x=t, then the system of equations will act as though y changes [|with respect to]x rather than t.

<span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">When graphing with sine and cosine, there are several noticeable patterns: <span style="color: #604a7b; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: 115%;">When graphing x=acos(t) and y=bsin(t), the system is a [|circle]. If x=a(cos(t))^2 and y=b(sin(t))^2 are graphed on a plane, they appear as a [|linear system]. The systems for sine and cosine to the third, fourth, fifth, and sixth powers (as shown above) have graphs similar to those of the function 1/x.

<span style="color: #482965; font-family: 'Tahoma','sans-serif'; font-size: 16pt;">﻿﻿ Click here for more information about patterns in parametric equations.

<span style="color: #482965; font-family: 'Tahoma','sans-serif'; font-size: 16pt;"> <span style="color: #482965; font-family: 'Tahoma','sans-serif'; font-size: 16pt;"> __<span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 22pt;">Real-World Applications __ <span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">Parametric equations are often used when tracking the position, velocity, or acceleration of a moving object. They are also used heavily in Computer-Aided Design and Drafting (CADD), when drafting designs on a computer. For more information about CADD, click here.

<span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">Most famously, parametric equations were used by Ptolemy in the 2nd Century AD to decide that planets travel in an epicycle.

<span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 16pt;"> __<span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 24pt;">History __

<span style="color: #5f497a; font-family: 'Tahoma','sans-serif'; font-size: 16pt; line-height: normal; margin: 0in 0in 10pt;">Although Ptolemy used a type of parametric equation system in the early centuries AD, they were first commonly used by Greek mathematicians. Descartes used them in the 1600s to examine spirals, and Leonhard Euler used them in 1745 to study the cycloid and write proofs about the cycloid.

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