Parametric Equations To Rectangular Form

Then far, the graphs we have drawn are divers past ane equation: a function with two variables, x and y . In some cases, though, information technology is useful to introduce a 3rd variable, called a parameter, and express x and y in terms of the parameter. This results in 2 equations, chosen parametric equations.

Let f and k be continuous functions (functions whose graphs are unbroken curves) of the variable t . Let f (t) = x and g(t) = y . These equations are parametric equations, t is the parameter, and the points (f (t), g(t)) make up a aeroplane curve. The parameter t must be restricted to a certain interval over which the functions f and g are defined.

The parameter tin have positive and negative values. Usually a plane curve is drawn as the value of the parameter increases. The direction of the plane curve as the parameter increases is called the orientation of the curve. The orientation of a plane curve can be represented by arrows drawn along the curve. Examine the graph below. It is defined by the parametric equations 10 = cos(t), y = sin(t), 0≤t < 2Π .

Figure %: A plane curve defined past the parametric equations x = cos(t), y = sin(t), 0 < t≤2Π .

The curve is the aforementioned one defined past the rectangular equation x ⁱⁱ + y ² = 1. It is the unit of measurement circle. Check the values of x and y at primal points like t = , Π , and . Note the orientation of the curve: counterclockwise.

The unit circumvolve is an example of a curve that tin can be hands fatigued using parametric equations. One of the advantages of parametric equations is that they tin can be used to graph curves that are not functions, similar the unit circle.

Another advantage of parametric equations is that the parameter can be used to represent something useful and therefore provide us with additional data nearly the graph. Often a plane curve is used to trace the motion of an object over a certain interval of time. Let'south say that the position of a particle is given by the equations from to a higher place, ten = cos(t), y = sin(t), 0 < t≤2Π , where t is time in seconds. The initial position of the particle (when t = 0)is (cos(0), sin(0)) = (1, 0). By plugging in the number of seconds for t , the position of the particle can be establish at any time between 0 and twoΠ seconds. Information like this could not be found if all that was known was the rectangular equation for the path of the particle, ten ⁱⁱ + y ⁱⁱ = 1.

It is useful to be able to convert between rectangular equations and parametric equations. Converting from rectangular to parametric can exist complicated, and requires some creativity. Here we'll hash out how to convert from parametric to rectangular equations.

The process for converting parametric equations to a rectangular equation is commonly called eliminating the parameter. Starting time, you must solve for the parameter in one equation. And so, substitute the rectangular expression for the parameter in the other equation, and simplify. Study the example below, in which the parametric equations x = twot - 4, y = t + 1, - âàû < t < âàû are converted into a rectangular equation.

parametric

t =

y =

+ 1

y =

x + 3

By solving for the parameter in 1 parametric equation and substituting in the other parametric equation, the equivalent rectangular equation was found.

1 thing to annotation about parametric equations is that more one pair of parametric equations can stand for the same plane bend. Sometimes the orientation is dissimilar, and sometimes the starting point is different, but the graph may remain the same. When the parameter is time, different parametric equations can be used to trace the same curve at unlike speeds, for example.