Problem 24 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The rectangular equation is \( y = - \left( \frac{x}{2} \right)^{2/3} \) with domain \( x \in (-\infty, \infty) \).

Step 1: Solve for Parameter t in Terms of x

We are given the parametric equations \( x = 2t^3 \) and \( y = -t^2 \). First, solve for \( t \) in terms of \( x \). From \( x = 2t^3 \), divide both sides by 2 to get \( t^3 = \frac{x}{2} \). Then take the cube root of both sides: \( t = \sqrt[3]{\frac{x}{2}} \).

Step 2: Substitute t into the y Equation

Substitute \( t = \sqrt[3]{\frac{x}{2}} \) into the second parametric equation \( y = -t^2 \). We have \( y = - \left( \sqrt[3]{\frac{x}{2}} \right)^2 \).

Step 3: Simplify the Equation

The expression \( \left( \sqrt[3]{\frac{x}{2}} \right)^2 \) is \( \left( \frac{x}{2} \right)^{2/3} \). So, the equation becomes \( y = - \left( \frac{x}{2} \right)^{2/3} \).

Step 4: Determine the Domain for x

Since the original equation for \( x \) implies that \( x = 2t^3 \) for all real \( t \), \( x \) can take any real value. Therefore, the interval for \( x \) is \( (-\infty, \infty) \).

Key Concepts

Rectangular Equations

In mathematics, equations can be of different forms, with rectangular equations being one of them. A rectangular equation is an expression that uses standard Cartesian coordinates, typically represented as \( x \) and \( y \). These equations describe the relation between these two variables in the two-dimensional plane.
To convert parametric equations into a rectangular form, we need to eliminate the parameter, often denoted as \( t \). This involves expressing \( t \) in terms of either \( x \) or \( y \), and substituting back into the other equation.

• This process helps transition from a parametrically defined path to a direct relationship between \( x \) and \( y \).
• In the provided solution, the parametric equations \( x = 2t^3 \) and \( y = -t^2 \) were successfully converted to the rectangular equation \( y = -\left(\frac{x}{2}\right)^{2/3} \).

Understanding rectangular equations is crucial because they are commonly used in graphing and analyzing functions geometrically in the Cartesian plane.

Domain of Functions

When dealing with functions in any form, determining the domain is essential. The domain of a function is the complete set of all possible input values (usually \( x \) values) for which the function is defined.
For the rectangular equation \( y = -\left(\frac{x}{2}\right)^{2/3} \), understanding the domain involves observing any restrictions that might limit \( x \).

• In this scenario, as \( x \) was defined by \( x = 2t^3 \) where \( t \) is any real number, \( x \) can also be any real number.
• This leads to the conclusion that the domain of the function is \( (-\infty, \infty) \).

Domains can change when different mathematical operations like square roots are involved, but cube roots and other operations in this function impose no limits, allowing \( x \) to freely range without constraints.

Cube Root Function

The cube root function is a relatively straightforward concept but manifests unique properties in equations. Represented generally as \( \sqrt[3]{x} \), it refers to finding a number that, when raised to the power of three, equals \( x \).
This function is distinct because the cube root of any real number exists and is a real number, including negative numbers. This differs from square roots, which don't exist for negative numbers within the real number system.

• In our exercise, solving for \( t \) in terms of \( x \) required taking the cube root, resulting in \( t = \sqrt[3]{\frac{x}{2}} \).
• Since the cube root function can handle negative inputs without issue, it poses no restriction on \( x \), reinforcing our full real number domain.

The ability to work with all real numbers makes the cube root function versatile for defining and understanding different types of functions presented in rectangular form.