Problem 68 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=x^{2}-3$$

Step-by-Step Solution

Answer

The parametric equation for (a) \(t = x\) is \(x = t\), \(y = t^2 - 3\) while that for (b) \(t = 2 - x\) is \(x = 2 - t\), \(y = (2 - t)^2 - 3\).

Step 1: Substitute \(t\) for \(x\) in Rectangular Equation

Start by substituting \(t\) for \(x\) in the equation \(y = x^2 - 3\) to get our parametric equations with parameter \(t\). For part (a), where \(t = x\), substituting \(t\) for \(x\) in \(y = x^2 - 3\) gives \(y = t^2 - 3\), thus obtaining the parametric equations: \(x = t\), \(y = t^2 - 3\).

Step 2: Substitute \(2 - t\) for \(x\) in Rectangular Equation

Next, for part (b), where \(t = 2 - x\), substituting \(2 - t\) for \(x\) into \(y = x^2 - 3\) to gives \(y = (2 - t)^2 - 3\). Our parametric equation set hence become \(x = 2 - t\), \(y = (2 - t)^2 - 3\).

Key Concepts

Rectangular to Parametric Conversion

Converting a rectangular equation to a set of parametric equations involves choosing a parameter, often denoted as 't', and expressing the coordinates, usually x and y, in terms of that parameter. Take the equation
\(y = x^2 - 3\)
as an example. To convert it, a common approach is to let the parameter 't' represent 'x', making the conversion straightforward.

By selecting
\(t = x\),
you can directly replace x with t in the rectangular equation to find the y-component of the parametric form:
\(y = t^2 - 3\).
Therefore, the pair of parametric equations will be:
\(x = t\),
\(y = t^2 - 3\).

This expresses the original curve in terms of a single variable t, which can represent any real number. As t varies, the values of x and y vary, tracing out the curve on the Cartesian plane. This technique is essential when graphing complex shapes or creating animations because it simplifies the motion to a single variable's flow.

Parametric Equations Representation

The representation of curves using parametric equations provides a vast advantage compared to the traditional rectangular (Cartesian) form. In parametric equations, each coordinate is a function of a single parameter. Considering the exercise provided:

\(x = t\) and \(y = t^2 - 3\),
the variable 't' can take on any real number, allowing for continuous representation of the point along the curve in the xy-plane. This form is especially beneficial for representing motion where time can be the parameter, showing the position of a moving point at any given time.

Parametric equations also allow one to capture more intricate curves that might be difficult or impossible to describe using the rectangular equation alone. Think of the curves on a roller coaster or the flight path of a soccer ball through the air; these can be meticulously detailed using parametric equations.

Algebraic Substitution in Equations

Algebraic substitution is a powerful tool in the transformation of equations. In our scenario, substitution is used to manipulate a rectangular equation into its parametric form. When the value of 't' is set as \(t = 2 - x\), one can perform a substitution to reformulate the y-equation:

\[y = (2 - t)^2 - 3\].
Here, algebraic substitution demands expanding the squared term and simplifying to ensure the resulting parameterization accurately represents the original curve. What we end up with is a new pair of parametric equations:
\(x = 2 - t\) and \(y = (2 - t)^2 - 3\),
which offers a different parametric perspective of the same equation. The exercise illustrates how, via algebraic substitution, various parametric equations can yield the same curve, providing flexibility in how one chooses to navigate the curve either for calculations or graphical representation.