Change Parametric Equation To Rectangular

So far, the graphs we have drawn are defined by one equation: a part with two variables, x and y . In some cases, though, information technology is useful to innovate a third variable, called a parameter, and express x and y in terms of the parameter. This results in ii equations, chosen parametric equations.

Let f and g exist 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), 1000(t)) make up a plane curve. The parameter t must be restricted to a certain interval over which the functions f and g are defined.

The parameter can have positive and negative values. Usually a plane bend is drawn as the value of the parameter increases. The management 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 divers past the parametric equations 10 = cos(t), y = sin(t), 0≤t < twoΠ .

Figure %: A plane curve divers by the parametric equations x = cos(t), y = sin(t), 0 < t≤iiΠ .

The bend is the same ane divers by the rectangular equation x ⁱⁱ + y ² = i. It is the unit circumvolve. Cheque the values of ten and y at key points like t = , Π , and . Note the orientation of the bend: counterclockwise.

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

Some other reward of parametric equations is that the parameter can exist used to represent something useful and therefore provide us with additional information about the graph. Often a airplane bend is used to trace the motion of an object over a certain interval of fourth dimension. Let'due south say that the position of a particle is given by the equations from to a higher place, 10 = cos(t), y = sin(t), 0 < t≤iiΠ , where t is time in seconds. The initial position of the particle (when t = 0)is (cos(0), sin(0)) = (1, 0). Past plugging in the number of seconds for t , the position of the particle tin can be found at whatever time betwixt 0 and 2Π seconds. Data similar this could not be constitute if all that was known was the rectangular equation for the path of the particle, x ² + y ^two = 1.

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

The process for converting parametric equations to a rectangular equation is unremarkably called eliminating the parameter. Start, you must solve for the parameter in i equation. Then, substitute the rectangular expression for the parameter in the other equation, and simplify. Study the example beneath, in which the parametric equations x = twot - 4, y = t + one, - âàû < t < âàû are converted into a rectangular equation.

parametric

t =

y =

+ one

y =

ten + 3

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

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