Problem 28 - Calculus Volume 3

Question

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\sqrt{t}, \quad y=2 t+4 $$

Step-by-Step Solution

Answer

The rectangular form is \( y = 2x^2 + 4 \) with domain \( x \geq 0 \).

Step 1: Understand the Relationship between x and t

The parametric equation for x is given by \( x = \sqrt{t} \). To express \( t \) in terms of \( x \), we square both sides to get \( t = x^2 \).

Step 2: Substitute t into y Equation

We know \( y = 2t + 4 \). Substitute \( t = x^2 \) into the equation for \( y \) to convert it into rectangular form. This gives us \( y = 2(x^2) + 4 \) or \( y = 2x^2 + 4 \).

Step 3: Determine the Domain for x

Since \( x = \sqrt{t} \), \( t \) must be non-negative (\( t \geq 0 \)). Thus, \( \sqrt{t} \geq 0 \), which implies \( x \geq 0 \). Therefore, the domain of the rectangular form is \( x \geq 0 \).

Key Concepts

Rectangular Form

When dealing with parametric equations, a common task is to convert them into a rectangular form. This involves expressing the relationships between variables in a classic cartesian coordinate system, using only the coordinates directly, without any parameters.

In this exercise, we start with two parametric equations, one for each axis: the x-equation is given by \( x = \sqrt{t} \) and the y-equation by \( y = 2t + 4 \). The goal is to eliminate the parameter \( t \) to find a direct relationship between x and y. This simplifies the expression of the curve in the Cartesian coordinate system.

• The first step is to express \( t \) in terms of \( x \). Since \( x = \sqrt{t} \), we can square both sides to obtain \( t = x^2 \).
• Next, we substitute \( t = x^2 \) into the y-equation \( y = 2t + 4 \). This yields \( y = 2(x^2) + 4 \), simplifying to \( y = 2x^2 + 4 \).

Thus, the rectangular form of the curve represented by the original parametric equations is \( y = 2x^2 + 4 \). This equation now solely describes the behavior of the curve in terms of x and y, providing a clear, direct algebraic relationship.

Domain of Functions

The domain of a function specifies the set of possible input values, which in turn determines the range of possible outputs. For our rectangular form equation, \( y = 2x^2 + 4 \), understanding the domain involves looking back at our parametric conversion.

We initially had \( x = \sqrt{t} \), meaning that \( t \) must be non-negative, as the square root function is only defined for non-negative numbers.

• This implies that \( \sqrt{t} \geq 0 \), which determines that \( x \geq 0 \).

So, the domain of the rectangular form equation is \( x \geq 0 \). This mathematical constraint reflects the fact that our curve exists in its defined form only when x is zero or positive. This consideration is essential for accurately describing the behavior of functions defined by parametric equations.

Algebraic Manipulation

Algebraic manipulation is a critical process in mathematics used to transform and simplify expressions. In converting parametric equations to rectangular form, manipulating the algebraic expressions enables solving for desired variables.

In this task, algebraic manipulation serves two main purposes:

• First, it allows you to isolate \( t \) in the x-equation. By squaring \( x = \sqrt{t} \), we rearrange it to \( t = x^2 \), removing the parameter from the equation so it can be substituted into the y-equation.
• Second, by substituting \( t = x^2 \) into \( y = 2t + 4 \), you directly express y in terms of x with \( y = 2x^2 + 4 \). This step reflects the capability of algebraic manipulation to relate the curve's characteristics in simple coordinate form.

The process of algebraic manipulation not only conveys the relationship between the necessary components of the equations but also clarifies their inherent connections and dependencies. In this way, algebra acts as a bridge between different mathematical forms, enhancing comprehension and application.