Rectangular equations are in the form of y = f(x), where the variables x and y represent the coordinates on a Cartesian plane. Parametric equations are in the form of x = f(t) and y = g(t), where the variable t represents the parameter. This means that parametric equations can represent a wider range of curves and shapes than rectangular equations.
To convert a rectangular equation to a parametric one, we can let x = t and solve for y in terms of t. This will give us the parametric equations x = t and y = f(t), where f(t) represents the solution for y in terms of t. We can also use the substitution method, where we substitute x = cos(t) and y = sin(t) into the rectangular equation to get the parametric equations.
Parametric equations are more useful than rectangular equations when dealing with curves and shapes that cannot be easily represented by a single rectangular equation. They are also useful for representing motion and transformations in physics and engineering, as the parameter t can represent time.
Yes, parametric equations can be graphed on a rectangular coordinate plane. However, the resulting graph may not be a single curve or shape, as the parameter t can represent different points on the curve at different values.
Parametric equations have several advantages over rectangular equations. They can represent a wider range of curves and shapes, they are useful for representing motion and transformations, and they can be graphed on a rectangular coordinate plane. They are also useful for solving difficult integrals and finding the length of curves.