Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=4 t^{3}, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The equation is \( y = \frac{4}{3\sqrt{3}} \cdot x^{3/2} \) with \( x \in [0, \infty) \).

Step 1: Express t in terms of x

We are given that \(x = 3t^{2}\). To express \(t\) in terms of \(x\), solve for \(t\):\[ x = 3t^{2} \]Divide both sides by 3:\[ t^{2} = \frac{x}{3} \]Take the square root of both sides to solve for \(t\):\[ t = \pm \sqrt{\frac{x}{3}} \]

Step 2: Express y in terms of x

Now, we have \(y = 4t^{3}\). Substitute \(t = \pm \sqrt{\frac{x}{3}}\) back into the equation for \(y\):\[ y = 4(\pm \sqrt{\frac{x}{3}})^{3} \]Since \((\pm)^{3} = \pm\), we only consider one sign:\[ y = 4 \cdot \left(\sqrt{\frac{x}{3}}\right)^3 \]Further simplifying gives:\[ y = 4 \cdot \left(\frac{x}{3}\right)^{3/2} \]

Step 3: Combine and simplify the equation

The expression \( y = 4 \left( \frac{x}{3} \right)^{3/2} \) becomes:\[ y = \frac{4}{3^{3/2}} \cdot x^{3/2} \]Simplifying \(3^{3/2}\) gives \(3 \cdot \sqrt{3}\), therefore:\[ y = \frac{4}{3\sqrt{3}} \cdot x^{3/2} \]

Step 4: State the interval for x

Since \(t\) is in \((-\infty, \infty)\) and \(x = 3t^{2}\), \(x\) will always be non-negative as it is a square term. Therefore, \(x\) is in the interval \[ [0, \infty) \]

Key Concepts

Parametric Equations

Parametric equations are a pair of equations that express the coordinates of the points on a curve as functions of a variable, often denoted as \( t \). This variable is typically referred to as the parameter. In this context, parametric equations offer a way of representing and linking geometric figures, like curves, through time or a progression of steps.
In the original exercise, we are given the parametric equations \( x = 3t^2 \) and \( y = 4t^3 \). These functions use \( t \) as the parameter to describe the path of a moving point on a particular curve as \( t \) varies over its interval, from \(-\infty\) to \(\infty\). As \( t \) changes, it generates corresponding values for \( x \) and \( y \), defining points on the curve within the coordinate plane.
Ultimately, the goal is often to eliminate the parameter and find a direct relationship between \( x \) and \( y \), which is known as the rectangular equation.

Interval Notation

Interval notation is a mathematical notation used to indicate a range of values, often real numbers. This is extremely useful in describing the set of inputs, or "domain", for which an expression is valid. The notation consists of numbers and brackets or parentheses, which denote whether endpoints are included or excluded in the interval.
In terms of our exercise, we have the interval for \( t \) defined as \((-\infty, \infty)\). This tells us that \( t \) can take any real number as its value, without any restrictions. However, when we substitute back into our rectangular equation, \( x = 3t^2 \), we observe that the squares of real numbers are never negative. Thus, \( x \) can only be non-negative, placing its interval at \([0, \infty)\).
This usage of interval notation helps clearly communicate which values are permissible for these variables, guiding the further analysis and solution of a given problem.

Square Root Functions

A square root function represents relationships where a variable is isolated by taking the square root of another expression. In our solution, we encountered the equation \( x = 3t^2 \) and sought to express \( t \) in terms of \( x \) by performing algebraic manipulations that involved the square root.
To solve for \( t \), we have \( t^2 = \frac{x}{3} \). By taking the square root of both sides, we get \( t = \pm \sqrt{\frac{x}{3}} \). The plus-minus symbol accounts for both the positive and negative square roots, indicating that \( t \) can be either positive or negative, reflecting two possible directions in interpreting the original path defined by \( t \).
Square root functions are prevalent in mathematical analysis and modeling, where we often deal with quantities related to physical dimensions or rates.

Algebraic Manipulation

Algebraic manipulation describes the process of rearranging and simplifying expressions to facilitate calculations or to achieve a desired form, such as solving for particular variables or eliminating parameters.
During the exercise solution, we performed several algebraic manipulations to transition from parametric to rectangular equations. Initially, from the parametric equation \( x = 3t^2 \), we isolated \( t \) as \( t = \pm \sqrt{\frac{x}{3}} \). This provided a means to substitute back into the second parametric equation, \( y = 4t^3 \).
Further manipulation involved substituting \( t \) to derive a function solely of \( x \) for \( y \), resulting in \( y = \frac{4}{3\sqrt{3}}x^{3/2} \). Such steps, which involve both algebraic simplification and substitutions, are essential for transitioning from parametric forms to more familiar Cartesian or rectangular notation.

• Isolate the desired variable.
• Simplify expressions using arithmetic operations.
• Use substitutions to connect relationships.

These techniques showcase the power of algebra in transforming complex relationships into more straightforward, interpretable forms.