Parametric Equations

In a cartesian plane, various equations are represented in simple algebraic form or using various complex functions like logarithmic, trigonometric, and much more. The parametric equations consist of various dependent variables and an independent variable represented using trigonometry. This helps solve these equations easily as the relationship between various variables is reflected. This study material notes on parametric equations provide various examples of curved surface representation on the cartesian plane by calculating trigonometric angles and other values.

Generating the Parametric Equations – Parameterization

In a graph, the representation of various curves is simplified by using parametric equations. These equations help generate or evaluate the value of a given function at various ranges so that the complete curve structure can be obtained. By evaluating the value of a graph at different ranges of coordinates, the quadrant of the figure can be obtained. The parametric representation of graph values includes some basic steps.

• Let’s take an equation: y = x² + 2, where m and n represent the x-axis and the y-axis coordinates, respectively.
• To find its parametric equation, take a new variable replacing the n, such that k = x + 2
• The values of x and k are interchangeable, and the final output will depend on the relation between x and k.
• This means, the value of k can be taken as k = x+ 2 or x = k + 2
• Finally, using the value of x = k + 2 in the given equation.
  • y = x² + 2
  • y = (k + 2)² + 2
  • y = k² + 4k + 6
• With this, at different values of k, various values of x and y can be found, and a graph can be plotted.

Parametric Equations for Higher-Order Derivatives

Parametric equations study material provides the steps and graph representation of complex equations, but there is a need to simplify these equations in case of differentiation. The single derivative of a complex equation can be found easily. Still, in real-life applications, there is a requirement for higher-order derivatives that requires the usage of a third variable. To illustrate the usage of parametric equations for higher-order derivatives, let’s take two equations, i.e., m = 2a and n = 3at + 4, where m, n, and t are variables and the rest are constants.

• To find dn/dm, the dependency of t is required.
• dn/dm = (dn/dt)*(dt/dm)
• Similarly, for finding​ d²n/dm^2,, the parametric equation for the given equations will be beneficial.
• Here, d²n/dm² = d/dm((dn/dt)*(dt/dm))
• d²n/dm² = (d/dt((dn/dt)*(dt/dm)))/(dt/dm)
• This can be further simplified by taking another variable, which would increase the number of steps and be less effective.
• The usage of t, i.e., intermediate variable for finding the relation between m and n, representing the x-axis and the y-axis coordinates, respectively, gives precise output in a short duration.
• Similarly, the higher-order derivatives of equations involving trigonometric functions used to represent curves in the cartesian plane can be obtained.

Parametric Equations for Curves – Circle

The representation of curves is one of the most interesting applications of parametric equations used to reduce the complexity of equations and increase efficiency while solving any equation. Understanding complex algebraic functions and calculus is required as the graph is designed using the trigonometric functions and proper sign values of the cartesian plane. The general representation for the equation of a circle of radius k is x² + y² = k², where x and y represent the coordinates of the x-axis and the y-axis, respectively.

• The differentiation of the equation w.r.t the radius, the x or the y coordinates is very complex as k = (x² + y²)*(1/2).
• The higher-order derivatives will be more complicated, hence, parametric equations are the best solution to reduce complexity.
• The parametric equation for representing a circle is x = k*cos(z) and y = k*sin(z) where z is the angle made by k, ie., radius, w.r.t the x-axis.
• These values help to get the higher-order derivatives of the circle by considering the intermediate value of z and trigonometric functions of cos(z) and sin(z).

Conclusion

Parametric equations are used to represent higher-order equations and curves in the cartesian plane. Using a parametric equation, the higher-order derivative and integral can be found in a minimum number of steps, and its complexity would be reduced. Hence, this representation allows multiple derivatives and integrals to generate precise values for a complex and varying equation.