Problem 12 - Precalculus with Calculus Previews

Question

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{3}+t+4, y=-2\left(t^{3}+t\right) $$

Step-by-Step Solution

Answer

The rectangular equation is \( y = 8 - 2x \).

Step 1: Write Down the Parametric Equations

Start by writing out the given parametric equations: \( x = t^3 + t + 4 \) and \( y = -2(t^3 + t) \).

Step 2: Express One Variable in Terms of t

To eliminate the parameter \( t \), we first solve one of the parametric equations for \( t \). Let's solve for \( t^3 + t \) from the \( y \) equation: \( y = -2(t^3 + t) \) can be rewritten as \( t^3 + t = -\frac{y}{2} \).

Step 3: Substitute Expression into Other Equation

Substitute \( t^3 + t = -\frac{y}{2} \) back into the equation for \( x \). Thus, \( x = -\frac{y}{2} + 4 \).

Step 4: Simplify the Rectangular Equation

The substitute results in the rectangular equation \( x = 4 - \frac{y}{2} \). You can rewrite this as \( y = 8 - 2x \) by solving for \( y \). This is the same graph in rectangular form.

Key Concepts

Rectangular Equation

A rectangular equation is a single equation in the Cartesian coordinate system that relates the variables typically seen as \( x \) and \( y \). These are also known as non-parametric equations, like the linear, quadratic, or any polynomial equation you might have encountered. Rectangular equations do not involve a parameter. Instead, they describe the relationship between \( x \) and \( y \) directly.

In the given exercise, we started with parametric equations where both \( x \) and \( y \) are expressed in terms of a third variable, \( t \). Our goal was to combine these into a single rectangular form. Instead of having \( x \) and \( y \) tied to \( t \), we aimed to write them in a single equation, like \( y = 8 - 2x \).

The purpose of seeking a rectangular equation is often simplification. In many cases, rectangular forms are more straightforward to graph and interpret, providing a traditional view of the relationships involved.

Parameter Elimination

Parameter elimination is the process of removing the parameter in parametric equations to derive a direct relationship between the variables \( x \) and \( y \). This is achieved through algebraic manipulation. You essentially solve one of the parametric equations for the parameter and substitute this expression into the other equation.

In this exercise, the parameter \( t \) was eliminated through the following steps:

• First, express one part of the equation (\( t^3 + t \)) in terms of \( y \), giving us \( t^3 + t = -\frac{y}{2} \).
• Next, substitute this expression into the other equation that involves the same expression (the \( x \) equation), resulting in \( x = 4 - \frac{y}{2} \).
• The result is a direct rectangular form \( x = 4 - \frac{y}{2} \), which can be further rearranged into \( y = 8 - 2x \).

Through parameter elimination, you streamline the representation of your equation, making it more straightforward to analyze or plot.

Graph Representation

Transforming parametric equations into a rectangular equation allows for easier graphing and visualization. In parametric form, you rely on a third variable, \( t \), to determine points on a graph for \( x \) and \( y \), typically generating more complex paths or shapes, such as circles or ellipses.

By converting to rectangular form, you directly plot \( y \) against \( x \). For example, the final rectangular equation \( y = 8 - 2x \) is a simple linear equation representing a straight line. This equation straightforwardly indicates a line with:

• Y-intercept at 8, meaning the line crosses the y-axis at the point (0, 8).
• A slope of -2, indicating the line falls 2 units vertically for every 1 unit it moves horizontally to the right.

Graphing \( y = 8 - 2x \) provides a clear visual of how \( x \) and \( y \) are interrelated over any interval. The linear nature of the graph is often easier to interpret and understand compared to its parametric counterpart.