Problem 69 - Precalculus

Question

Find a set of parametric equations to represent the graph of the rectangular equation using (a) \(t=x\) and \((b) t=2-x\) $$y=3 x^{2}+1$$

Step-by-Step Solution

Answer

Parametric equations using \(t=x\) are \(x=t\) and \(y=3t^2+1\). Parametric equations using \(t=2-x\) are \(x=2-t\) and \(y=3(2-t)^2+1\).

Step 1: Understanding Rectangular and Parametric Equations

A rectangular equation is what we're most used to, where we express one variable in terms of another (like \(y=3 x^{2}+1\)). A parametric equation, on the other hand, expresses each variable as a function of one parameter. Our task is to rewrite the given rectangular equation into two parametric ones, using the provided parameters.

Step 2: Applying the Parameter \(t=x\)

The first part of the exercise asks for the parametric equations using \(t=x\) as the parameter. To obtain the parametric representation, express x and y in terms of the parameter, which in this case is \(t=x\). For x, the parametric equation is simply \(x=t\). Then, substitute \(t=x\) into the given equation to get the parametric equation for y. That is, replace every instance of \(x\) in the original equation with \(t\) to get \(y=3t^2+1\).

Step 3: Applying the Parameter \(t=2-x\)

The second part of the exercise asks for the parametric equations using \(t=2-x\) as the parameter. This is relatively more complex than the first part. Rewrite \(t=2-x\) to express x in terms of t, resulting in \(x=2-t\). Then, substitute \(x=2-t\) into the given equation in place of \(x\). This gives us \(y=3(2-t)^2+1\).

Key Concepts

Rectangular Equation

When first learning about equations in algebra, you probably encountered the term 'rectangular equation'. This formal term refers to the more common equations that plot out curves or lines on the Cartesian coordinate system, which is the graph with x and y axes. An example of a rectangular equation is the given expression \(y = 3x^2 + 1\). It neatly describes a relationship between y and x, such that for every x value there is one corresponding y value. This kind of equation is labeled 'rectangular' because each axis in the Cartesian system is perpendicular to the other, forming right, or rectangular, angles.

Understanding rectangular equations is crucial because they give a direct relationship between variables that we can use for graphing or solving problems. Although the rectangular form provides a straightforward approach, it does sometimes limit how we think about and manipulate the relationships between different mathematical quantities.

Parametric Representation

Parametric equations offer a different perspective by introducing a 'parameter', which is a third variable that both x and y depend on. It's like introducing a middleman that helps x and y communicate. Instead of relating x and y directly, they both depend on a new variable, usually t.

Take for example the parametric equations derived from the rectangular equation \(y = 3x^2 + 1\) using the parameter \(t = x\). The x-component of the parametric representation is \(x = t\), and the y-component is \(y = 3t^2 + 1\) after substituting t for x in the original equation. This gives us a new way to describe the curve: as t changes, we can trace how both x and y change accordingly. This format is quite powerful, especially in fields like physics where time often plays the role of the parameter t, dictating the position of objects in motion.

Algebraic Manipulation

Algebraic manipulation is the process of transforming mathematical expressions into different equivalent forms using algebraic rules. This skill is essential in converting rectangular equations to parametric form, among many other applications. It involves operations such as adding, subtracting, factoring, expanding, and substituting.

For instance, when given the parameter \(t = 2 - x\), algebraic manipulation is required to express x in terms of t, flipping the equation to get \(x = 2 - t\). Then, we substitute this expression into the original rectangular equation. This substitution leads to further manipulation: expanding and simplifying. We end up with the parametric equation \(y = 3(2 - t)^2 + 1\), which now expresses y in terms of our parameter t. This kind of manipulation is the bread and butter of algebra and will be a skill you rely on throughout your studies in mathematics.