Problem 50 - 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) \(t=2-x.\) $$y=x^{3}-x^{2}$$

Step-by-Step Solution

Answer

For the parameter \(t = x\), the set of parametric equations will be: \(x = t\) and \(y = t^{3} - t^{2}\). For the parameter \(t = 2 - x\), the set of parametric equations will be: \(x = 2 - t\) and \(y = (2 - t)^{3} - (2 - t)^{2}\)

Step 1: Evaluate for the parameter \(t=x\)

We let \(t=x\) in the given equation. This gives us \(x = t\) and \(y = t^{3} - t^{2}\). Therefore, when \(t = x\), the set of parametric equations becomes: \(x = t\) and \(y = t^{3} - t^{2}\)

Step 2: Evaluate for the parameter \(t = 2 - x\)

Now, we let \(t = 2 - x\) in the given equation. Therefore, we solve for \(x\) in terms of \(t\), which gives us: \(x = 2 - t\). Substituting this value into the function for \(y\), we get: \(y = (2 - t)^{3} - (2 - t)^{2}\). Therefore, when \(t = 2 - x\), the set of parametric equations becomes: \(x = 2 - t\) and \(y = (2 - t)^{3} - (2 - t)^{2}\)

Key Concepts

Rectangular to Parametric Conversion

Transforming a rectangular equation into parametric equations is a process that can bring a lot of flexibility to how we describe curves on a plane. In our scenario, the rectangular equation provided is
\[ y = x^3 - x^2 \]
To convert this into a parametric form, we'll use a parameter, typically denoted as 't', to represent the variables x and y independently. Let's discuss the two suggested parameters:

• For (a) \(t = x\), what we're essentially doing is letting another variable, t, take on the values that x would. This is a direct substitution which yields the parametric equations:
  \[ \begin{align*} x &= t \ y &= t^3 - t^2 \end{align*} \]
• For (b) \(t = 2 - x\), we're assigning a relationship between t and x, which after solving for x gives us:
  \[ x = 2 - t \] This transformation allows us to express y in terms of t as well, resulting in:
  \[ y = (2 - t)^3 - (2 - t)^2 \]

Each of these two sets of equations allows us to trace the same curve originally represented by the rectangular equation, just through different lenses.

Parametric Representation of Functions

Parametric representation of functions introduces a third variable, a parameter, that usually offers greater control over the description of a graphical representation. When dealing with parametric equations, we are provided a way to express each axis independently in terms of another variable.
In describing curves, this can be extremely useful for complex motion descriptions or when the relationship between x and y is not straightforward or is cumbersome to manipulate algebraically.
In the given exercise, we have translated a cubic function into such a form, which has key benefits such as:

• Easier calculation of derivatives and integrals in certain cases
• The ability to describe multiple values of y for a single x value without ambiguity
• The inclusion of a 'time' factor if the parameter t is considered to represent time, allowing us to describe dynamic systems

Parametric equations provide a new perspective and, in some cases, a more natural way to describe phenomena in mathematics and physics.

Algebraic Manipulation

Algebraic manipulation plays a crucial role when working with parameterization as it allows us to express and solve equations in terms of the newly introduced parameters. It often involves rearranging terms, factoring, expanding, and applying algebraic identities to achieve a desired form.
In our example with \(t = 2 - x\), after replacing x in the rectangular equation, we perform the necessary expansions and simplifications to express y:
\[ y = (2 - t)^3 - (2 - t)^2 \]
Expanding these terms requires an application of the binomial theorem and then combining like terms. The algebraic skills used here are fundamental for students as they foster an understanding of how different forms of an equation can equivalently represent the same relationship and how we can transition between those forms efficiently.
Additionally, masterful algebraic manipulation is essential for calculus, physics, and higher-level mathematics, as it underpins the ability to work with complex expressions and to simplify problems to make them more tractable.