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Graphing calculators and computer algebra systems sketch a plane curve by plotting points corresponding to a large number of values of the parameter t and then connecting the plotted points with a curve.
The general setup to imagine is pic-tured: An object moving around a circle of radius cen-tered at a point in the -plane. The path traced out is the circle. However, the location of the object at time will depend on a number of things: The starting location of the object; Figure 22.11: Circular mo-tion.
Practice 1: Find parametric equations for the lines through the point P = (3,–1) that are (a) parallel to the vector A = 〈 2, –4 〉 , and (b) parallel to the vector B = 〈 1, 5 〉 . Then graph the two lines. The parametric pattern works for lines in three dimensions.
Parametric Equations of Lines on a Plane. = 4 – 2t. = 5 + 3t. Use a table of values with three values of t to plot the graph. Eliminate the parameter to find an EXPLICIT equation for y as a function of x. Solve for t in terms of x. Substitute into the y equation to eliminate t. x.
There are several basic examples of parametric equations that are good to know: (1) Graphs of functions. Using parametric equations is a true generalization of the y= f(x)
In another word, a parametric curve is two maps from I to R, which are x(t) and y(t). (Here the interval I can be (0,1), [0,2], [0,2⇡), R and so on. ) Example 1.3. (1) Graph the parametric curve r(t)=(x(t),y(t)) = (cost,sint),t2 [0,2⇡]. (2) Graph the parametric curve r(t)=(x(t),y(t)) = (cos2t,sin2t),t2 [0,2⇡].
Parametric equations describe the location of a point (x,y) on a graph or path as a function of a single independent variable t, a "parameter" often representing time. In 2 dimensions, the coordinates x and y are functions of the variable t: x = x(t) and y = y(t) (Fig. 5).
