Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t, y=t-1, \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \( y = \frac{x}{3} - 1 \) with \( x \) in \( (-\infty, \infty) \).

Step 1: Express t in terms of x

Given the parameterized equation for \( x \), \( x = 3t \), solve for \( t \) by dividing both sides by 3. Thus, \( t = \frac{x}{3} \).

Step 2: Substitute t in y equation

Using the expression from Step 1, substitute \( t = \frac{x}{3} \) into the equation \( y = t - 1 \). This gives us \( y = \frac{x}{3} - 1 \).

Step 3: Simplify to find the rectangular equation

After substitution, you have the equation \( y = \frac{x}{3} - 1 \). Simplify this equation to express \( y \) in terms of \( x \): \( y = \frac{x}{3} - 1 \).

Step 4: Determine the interval for x

Since \( t \) can take any real number value from \( (-\infty, \infty) \), and \( x = 3t \), \( x \) can also take any real number value. Therefore, the interval for \( x \) is \( (-\infty, \infty) \).

Key Concepts

Parameterized Equations

Parameterized equations are expressions where one or more variables are expressed in terms of a third variable, called a parameter. In our exercise, the equations for both \(x\) and \(y\) are expressed in terms of \(t\), which is the parameter.

• For \(x\), we have \(x = 3t\).
• For \(y\), we have \(y = t - 1\).

By adjusting \(t\), we can generate pairs of \((x, y)\) coordinates, describing a path or curve. Such parameterization is especially useful in dealing with curves that are not easily described by a single equation in rectangular form. Converting these into rectangular form involves eliminating the parameter, allowing us to see the relation directly between \(x\) and \(y\). Understanding how to convert parameterized equations into rectangular form is essential for interpreting the geometry of these relations.

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are used to specify the position of a point in a plane using two numbers, \(x\) and \(y\). The point \((x, y)\) describes a small position in the two-dimensional space defined by the perpendicular \(x\) and \(y\) axes.
In our example, we have derived a rectangular equation from the parameterization. We started with the parameterized form \(x = 3t\) and \(y = t - 1\), and then eliminated \(t\) to get \(y = \frac{x}{3} - 1\). This equation now directly relates \(x\) and \(y\), allowing us to plot the line without relying on \(t\). This process is important as it provides a clearer geometric interpretation, enabling easier analysis and graphing of the relation.

Interval Notation

Interval notation is a standardized way to express the set of numbers between two endpoints. In our problem, we use interval notation to express the range of possible values for \(x\).
From the solution, since \(t\) can be any real number \((-\infty, \infty)\), and \(x = 3t\), \(x\) also spans all real numbers. Therefore, the interval for \(x\) is \((-\infty, \infty)\).

• \((-\infty, \infty)\) signifies all real numbers without restriction.
• Square brackets \([a, b]\) imply inclusion of endpoints, while round brackets \((a, b)\) imply exclusion of endpoints.

Understanding and using interval notation is essential for clearly communicating the constraints of your function or relation. It allows mathematicians to specify ranges of input or output values neatly and concisely.