Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

Here's a brief answer to the question. For \(t=x\), the parametric equations are \(x=t\) and \(y=\frac{1}{t^{2}}\), and for \(t=2-x\), the parametric equations are \(x=2-t\) and \(y=\frac{1}{(2-t)^{2}}\).

Step 1: Using Parameter \(t = x\)

We start by replacing \(x\) with \(t\). So, the parameterized equations become \(x=t\) and \(y=\frac{1}{t^{2}}\). So, we have our parameterized equations with \(t=x\).

Step 2: Using Parameter \(t = 2-x\)

Next, we follow the same procedure but replace \(x\) with \(2-t\) to accommodate our second parameter. Thus, \(x=2-t\), and substituting \(x\) by \(t\) in the equation \(y=\frac{1}{x^{2}}\), we get \(y=\frac{1}{(2-t)^{2}}\). This gives us the parametric equations for this case.

Key Concepts

Rectangular Equation

A rectangular equation is a mathematical expression that relates two variables, typically \(x\) and \(y\), using algebraic operations. In our problem, the rectangular equation provided is \(y = \frac{1}{x^2}\). This is a rational function where \(y\) is expressed directly in terms of \(x\). Such equations are often graphed on the Cartesian coordinate system, which uses perpendicular \(x\) and \(y\) axes.

Understanding this form is crucial as it serves as a basis for converting into parametric equations, which can offer more flexibility and insights when describing curves. Converting a rectangular equation into parametric form involves introducing a third variable, a parameter, that expresses both \(x\) and \(y\). This transformation is beneficial for complex curves, allowing for clear graphical representation and simplified calculations.

Parameterization

Parameterization involves expressing a curve by defining both \(x\) and \(y\) in terms of a new variable \(t\), known as the parameter. This process helps in describing curves that are otherwise challenging to express using conventional rectangular equations.

In our example, two parameterizations were explored:

• For the first, letting \(t = x\), we directly replace \(x\) with \(t\), giving us the set: - \(x = t\) - \(y = \frac{1}{t^2}\)
• For the second, using the transformation \(t = 2 - x\), we replace \(x\) with \(2-t\): - \(x = 2 - t\) - \(y = \frac{1}{(2-t)^2}\)

By changing parameters, we effectively reframe how we view and plot the original equation, providing alternative perspectives and simplifying the depiction of the function.

Graph Representation

Graph representation is an essential part of visualizing mathematical equations. By plotting equations, we can gain insights into the behavior of functions and the relationships between variables. For the original rectangular equation \(y = \frac{1}{x^2}\), the graph is a curve representing points where this equation holds true.

By converting the rectangular equation into parametric form, each parameterization yields a different approach to plotting the same curve:

• Using \(t = x\), the curve is traced as \(t\) varies, respecting the original relationship between \(x\) and \(y\).
• With \(t = 2-x\), the same curve is covered differently, starting from a distinct point or direction.

Parametric equations thus provide a robust framework for exploring complex paths and intersections that are otherwise cumbersome to describe.

Mathematics Education

Understanding parametric equations is a significant aspect of mathematics education. This concept is highly applicable in fields such as engineering, physics, and computer graphics, where describing motion or trends requires flexibility and precision.

Educationally, parameterization helps students grasp how equations can change under transformation and teaches them to view problems from multiple angles. It also reinforces their understanding of linear algebra and calculus, as many real-life applications involve these equations. Learning how to switch between rectangular and parametric forms fosters a deeper comprehension of geometry and functional analysis.

Activities that include converting equations, exploring their graphs, and analyzing distinct parameterizations promote critical thinking and problem-solving skills among students.